Certainly, Imam Ali’s (as) greatest concern would have been to continue the way of the Holy Prophet (saw) and guard the path of hidayah (guidance) and true Islam, so that mankind could reach sa’adah (salvation) both in this world and the next. The Holy Prophet of Islam (saw) has referred to this clear path as “المحجة البیضاء” and even those who somehow played a role in distancing the Imam from the caliphate never denied this fact.
The righteous and pious whose aim is to only obey God and obtain His satisfaction are true examples for this Quranic script: “Those who, should We establish them in the land, will keep up prayer and pay the zaka’t and enjoin good and forbid evil; and Allah's is the end of affairs”
To answer the question perhaps we can reverse the terms of the question by saying, what changes would’ve taken place in the world of Islam if the Holy Prophet (saw) had lived for another thirty years?!
The reason being that when the Prophet (saw) explicitly states that Ali’s (as) relation to him, is like the relation of Aaron (as) to Moses (as), and with keeping in mind that the term “انفسنا” of the verse of Mubahalah refers to Ali (as), according to the ijma’ (consensus) of the Muslims that no one was with the Prophet on that day (of Mubahalah) except for Ali (as), Fatima (sa) and their two sons; we can deduce that the Quran portrays Ali (as) to be the nafs (self, soul) of the Prophet (saw). This issue was repeated by the Prophet (saw) himself several times, using different expressions, one of which is: “Your being is from me and my being is from you”, etc., hence, there is no way one can imagine, whether Imam Ali was the caliph or marginalized and deprived of it, that his lifestyle would have been any different, after the death of Prophet (saw), than that of the Prophet’s (saw) clear and well-defined lifestyle.
Therefore, the short answer to this question is that if Imam Ali (as) had became caliph he would have continued and persisted in the path that the Prophet (saw) had formerly outlined throughout his life. For instance, he would have endeavored to keep the path of hidayah (guidance) clear and open to assist the seekers of salvation in reaching the ultimate destination of spiritual excellence.
There is an phrase in the Arabic language “المحجة البیضاء” which is translated to “the clear path”. The Holy Prophet (saw) addressed his companions and said: “If you obey Ali (as) after me – which I do not assume you will! – you’ll definitely find him to be both guided and a guider that will lead you to “the clear path”.”
In the beginning of Imam Ali’s (as) caliphate, after listening to one of his companions conveying the complaints of some of the Sahaba regarding his just distribution of the Bayt al-Maal (treasury), he said: “I swear by God, if my life continues and I stay immune of the evil (plans), I will lead the people to “the clear path”. Also, in an explanation of why the six-man council, did not select him as caliph, after the death of the second caliph, he states: “When they realized I bear nothing but the Book of Allah, “the clear path” and the will of the Prophet (saw) (and they were not able to reach their personal interests through me), they removed me from caliphate and passed it to another person.” It is interesting to know that even the caliphs that were somehow constantly preventing Imam Ali (as) from getting near the position of caliphate would not only never deny this truth – that Ali (as) would have lead the clear path – but even at times, they would verify it! In a private talk with Ibn Abbas, Umar Ibn Khattab testifies that: “If Imam Ali (as) becomes caliph after he (Umar) dies, he will definitely lead the people to “the clear path”.”
Additionally, when the second caliph was choosing the members of the council that was meant to assign the next caliph and after criticizing every member of having a deficiency, including Imam Ali for having a sense of humor, he asserted: “I swear to Allah that if your [Ali (as)] faith was placed on the scale and the faith of every person in the whole world was placed on the other side of the scale, surely your faith would prove to be heavier! Imam Ali (as) left the meeting after hearing these words, then Umar Ibn Khattab addressed the rest of the audience and said: I know someone who is capable of leading you to “the clear path” if you’d only submit yourselves to him! The people present in the meeting immediately asked: Who is he? Umar replied: The same person who just left…!
Based on the ahadith mentioned above, we can see how if caliphate had been given to Imam Ali (as) the most important thing he would have done was to keep the path of salvation clear and open to everyone so that there would be no excuse left for going astray. In line with this, he says that if the grounds existed I would convince the followers of the Torah, Injil (Bible), Zabur (the holy book of David (as)) and Quran to the right path based on their own holy books. He did not allot essential value to caliphate and merely sought it to be useful in order to spread and govern justice, otherwise, holding power was not even as significant to him as he himself puts it, as his sown up shoe or the sneeze of a goat! This was the mentality he would constantly advise his executives to take up, so they would perceive their positions and ranks as a trust entrusted to them by Allah, rather than food and prey (to run after and try to snatch from each other, like wolves). It was of little consequence to him how many lands and countries were conquered and added to his reign in the period of his caliphate, what he conceived to be important was to direct the people towards God and the hereafter by keeping the path of righteousness recognizable and open. We can extract this character of his from the short period he was caliph, because, despite the fact that he faced a variety of problems and had no choice but to participate in a series of battles, he was able to prevent mental and behavioral deviations, from distorting the true Islam. What is interesting is that the major lands conquered by preceding caliphs were Iran, Iraq, Syria and Egypt which are currently, either Shia affiliated countries or lovers of Imam Ali (as) and his progeny; proving that although Imam Ali (as) didn’t have a direct role in conquering these lands, God instilled the love of Imam Ali (as) in their hearts for this to be an example of the verse that says God instills the love of the good and righteous in the heart of the people.
The speech of Fatima (sa) in the mosque, is maybe the best answer to your question: “What made you criticize Ali (as)? You criticized him (Ali) because his sword did not distinguish between one who he knew and the stranger, nor between the brave and fearful. They realized that he is not afraid of death. They saw how he would attack them and send them to the next realm and would spare some, only to be dealt with by others or to be a lesson to the rest. He knew the book of Allah very well and his wrath was only seen for Allah’s satisfaction. I swear to Allah if they hadn’t deviated from the clear and open path of righteousness and hadn’t abstained from adhering to reason and sound logic, he would have guided them to the straight path and prevented them from going astray. I swear to Allah that if your men had not persisted in removing the ‘matter’ (of caliphate) from Ali’s hands and would have left it to him just like how the Prophet had handed it to him, he would have conducted it in a patient manner and escorted this camel to its final destination safe and sound. Ali would direct them to a broad, clear and overflowing water trough… . He would quench their thirst from this water.
Ali wishes the best for them, both in their presence and absence. If he would have taken the position of caliph he would never have consumed more than a little amount from the Bayt al-Maal, enough water to quench his thirst and enough food to relieve his hunger. (Only at that time) would the pious and evil be distinguished and the honest be separated from the hypocrite, then the doors of Allah’s blessings would open down onto us. Allah will soon punish them for their corrupt deeds.”
In the end, since your inquiry may be intrigued by a doubt raised by the enemies of Amir Al-Mu’mineen (as) themselves, who claim that all of the victories and accomplishments of the Islamic nation were obtained before the caliphate of Imam Ali (as) and it was only when he became caliph that problems began to arise and civil war broke out for the first time, we must say:
1. Civil war actually began when a group of Muslims fought with the third caliph in which resulted in his assassination; this chaos was what initiated all the predicaments throughout the period of Imam Ali’s (as) caliphate!
2. If Imam Ali (as) had allotted most of his time to expanding the Islamic sovereignty and stood in peace with the Naketheen, Qaseteen and Mariqeen and had given in to their demands, instead of abolishing their creed and diminishing them, surely there would not be a sign of true Islam and Islam would have transformed into a label used to cover for material desires.
 Hajj:41, “الَّذینَ إِنْ مَکَّنَّاهُمْ فِی الْأَرْضِ أَقامُوا الصَّلاةَ وَ آتَوُا الزَّکاةَ وَ أَمَرُوا بِالْمَعْرُوفِ وَ نَهَوْا عَنِ الْمُنْکَرِ وَ لِلَّهِ عاقِبَةُ الْأُمُور”.
 Sahih Bukhari, vol. 4, pg. 208, Dar al-Fikr, Beirut, 1401 (AH).
 Ale Imran:61, “فَمَنْ حَاجَّکَ فیهِ مِنْ بَعْدِ ما جاءَکَ مِنَ الْعِلْمِ فَقُلْ تَعالَوْا نَدْعُ أَبْناءَنا وَ أَبْناءَکُمْ وَ نِساءَنا وَ نِساءَکُمْ وَ أَنْفُسَنا وَ أَنْفُسَکُمْ ثُمَّ نَبْتَهِلْ فَنَجْعَلْ لَعْنَتَ اللَّهِ عَلَى الْکاذِبینَ”.
 Sahih Muslm, vol. 7, pp. 120-121, Dar al-Fikr, Beirut.
 Sahih Bukhari, vol. 4, pg. 207.
 Majlisi, Muhammad Baqir, Bihar al-Anwar, vol. 35, pg. 398, quoted by Hafez Abu Na’im, one of the great Sunni scholars, Al-Wafa’ Institute, Beirut, 1404 (AH).
 Ibn Abi al-Hadid, Sharh Nahjul-Balaghah, vol. 7, pg. 37, Library of Ayatullah Mar’ashi Najafi, Qom, 1404 (AH).
 Majlisi, Muhammad Baqir, Bihar al-Anwar, vol. 31, pg. 347.
 Ibid, pg. 417.
 Ibn Abi al-Hadid, Sharh Nahjul-Balaghah, vol. 12, pp. 259-260.
 Majlisi, Muhammad Baqir, Bihar al-Anwar, vol. 26, pg. 182, hadith 8 (tens of similar hadiths can be found in other soiurces).
 Nahjul-Balaghah, pg. 76, sermon 33, Dar al-Hijrah Publications, Qom.
 Ibid, pg. 49, sermon 3.
 Ibid, pg. 366, letter 5.
 Maryam:96, “ان الذین آمنوا و عملوا الصالحات سیجعل لهم الرحمن ودا”.
 Bihar al-Anwar al-Jami’ah li Durar Akhbaar al-A’immah al-At’haar, vol. 43, pg. 159; Zendegaaniye Hazrate Zahra Alayhassalam (Rowhani), pp. 581 and 582.