There are two questions here, one about the role of Constantine I, the other about the conversion of the empire itself, which are two quite separate issues. Constantine probably converted sometime as an adult; the exact time of conversion is debatable and we should probably see his conversion as a process rather than a specific choice made at a particular time. There are other views as well: Peter Heather for example prefer to see the future emperor as a Christian from the start who disguised his views until it became acceptable to reveal his personal faith. We really can't say anything for definitive on this and it is I think better to recognise that there is a debate, rather than only follow certain historians' arguments. What is more clear is Constantine's actions as the senior emperor, as he did favour Christians in his court. However even so Christianity was not made the state religion until the reign of Theodosius I in 380. Emperors between Constantine and Theodosius ranged from the pagan Julian, 'Arians' like Constantius II and Valens, as well Valentinian I, who didn't seem to have cared all that much what religion people followed. Constantine and his sons were all firm Christians (albeit of different beliefs, Constantine had flip-flopped on 'Arianism' and Constantius II had actively promoted this 'heresy'), but can we describe the empire they ruled as a Christian one? From certain perspectives, yes, since they built churches and involved themselves in church councils, but from other perspectives, no, since many prominent pagans remained in positions of power and influence, even after Theodosius' edicts.
It was certainly helpful for Christianity's growth that it was now a legal religion and often benefited from imperial patronage, but we should not forget that other factors were at work as well; it is ultimately an unanswerable question whether Christianity would have continued to prosper if the imperial elite did not (generally) follow its tenets, but I'm more inclined to note the importance of other contingent influences. Christianity was not just a religion, distinct from political or social issues (as we would perhaps think based on our experiences of religion in the present day), but instead a phenomenon that affected every aspect of life, so it is not clear what would have instead happened if we took away its political crutch. We can perhaps look at the accounts of persecutions for a sense of how Christians would react when they no longer had any political support, yet this approach also brings with it its own problems. Under the pagan Julian, Christians were not able to teach rhetoric and paganism, at least the form the emperor favoured, was promoted. Yet Julian's schemes came to nothing, with even his pagan revival being quite unpopular amongst the very people he was trying to appeal to. Many Christians were certainly upset that their now traditional privileges were taken away, but not all Christians thought the same - Donatist 'heretics' in North Africa for example had instead experienced a revival under Julian since the emperor sought to divide the Christian community by favouring previously neglected groups. This is a rather crucial point, since Christianity was by no means united and it is perhaps more useful to refer to it as many Christianities instead. We can also point to the religious policies of the 'Arian' Constantius II and Valens - they did promote the interests of some Christians, but they also persecuted others because of their 'heretical' beliefs. Evidently, Christianity as a whole did not always benefit from its close relationship with the state, since it can just as easily be used against it.
Another important point is that Christianity was already widespread even before Constantine. Estimates for Christians within the empire c.300 range from 3%-10% (the lower range being Peter Heather's figure, the higher percentage is from Rodney Stark), but it was nonetheless a tangible minority that had its adherents across the Roman world. Some historians even see Constantine's mother as a Christian before the emperor's conversion; this point is debatable, but what is certain is that Constantine's father, the Caesar Constantius I, was a pagan who nevertheless tried to rein in the persecution of Christians ordered by the senior emperor Diocletian. The same was true for Maxentius, who seized power in Rome in 306 and ended the persecution there. Lactantius, the famous Christian apologist for the Constantinian dynasty, had likewise once been invited to the court of Diocletian to teach rhetoric before the persecution began. Sympathy for Christianity was clearly in place even at the highest section of Roman society and Christians were likewise able to thrive in a pagan world. An extended study of this phenomenon can be found in D. Boin's Coming Out Christian in the Roman World: How the Followers of Jesus Made a Place in Caesar’s Empire (2015), as he explains very well how many Christians, both from a high social background and those from the very bottom, were perfectly integrated into traditional Roman society. Many others of course did not and chose to live a life that was at odds with Roman sensibilities, but so did many pagans. We should never generalise about complex social phenomena, such as the Roman attitude to Christianity, but instead recognise the diversity and divergent nature of people's reactions. People, as always, are very complicated creatures, so again I find it difficult to evaluate Constantine's personal role in the spread of Christianity.
Moreover, by the fourth century there were even vibrant Christian communities well beyond the empire, who surely would have continued to grow (or decline) regardless of Constantine's personal conversion:
The Syriac kingdom of Osrhoëne, focused on its main urban center of Edessa, probably was the first to embrace the new religion, perhaps as early as the reign of king Abgar VIII in the third century. Situated near the Euphrates, astride the principal caravan route between the Roman and Sassanian empires, Edessa served as a vital conduit for Christianity into the Persian east. The church flourished within Sassanian Mesopotamia until Constantine’s ill-considered letter to Shapur II, in which he recommended Christianity to the shahan’shah and voiced his concern for Shapur’s Christian subjects. Henceforth, Christians suffered repeated persecution within the Persian realm, although they continued as one of the Persian Empire’s most significant religious minorities. Christianity followed trade routes down the Red Sea, and indeed, strong evidence suggests that Christian communities were established along the Kerala coast in south India as early as the first or second centuries. Christianity also took root in the Caucasian regions of Colchis, Iberia, and Armenia, where king Tiridates IV converted to Christianity perhaps as much as a full decade before the Battle of the Milvian Bridge in 312.^1
From this perspective, it was perhaps not helpful for many Christians that Christianity became so identified with the Roman empire, since in doing so those in Persia were seen as potential traitors by their rulers, who were otherwise quite content to let them believe whatever they want. On the other hand, it could also promote the spread of Christianity, as it encouraged these Christians to travel far beyond imperial borders. In the seventh century for instance 'Nestorians' (more properly members of the Church of the East), long-condemned by the church within the Roman empire, reached China from Persia and were welcomed by the Chinese emperor; though they were now not welcome in the Mediterranean world, 'Nestorians' were nonetheless able to flourish in places far from it!
TL; DR: No idea. Counter-factual history is complicated.
Follow-up question : how followed was christianity when Constantine converted? Was is already the majority of the empire's religion? Conversely, what was the state of paganism?
Hi! Not to discourage further discussion, but /u/talondearg addressed a similar question a while back on the conversion of Constantine and the changing social position of Christianity during the fourth century.
If you have any follow-ups, make sure to ping the user in your response!
I always thought that emperor constantine decided to embrace christianity, as it was becoming a nuisance for the empire, since it was spreading ideals and beliefs that were sort of, anti-roman. By that I mean the warfare, spreading of rome, indulging in great wealth, taking slaves and all that sort of stuff. But also that christianity was an institution of control that the roman emperor could not influence.
So his move to embrace christianity was strategically brilliant, in that it motivated his troops, which appear to have already been largely christian, to fight for him, which won him the battle of the milvian bridge, but also put him into a position where he might begin to take control of christianity and use it to control his people in a similar way that earlier emperors would have used the pagan gods.
If that were a fair representation of events, then I would say christianity would have already been on the rise to an extent, and the move would have helped spread it, but it was also a sort of move for who controlled it.
In Nero's days, he blamed the christians for burning down that section of Rome where he later built his palace extension, so you can see that at that time christians were definitely not sort of "pro empire" and then in the medieval ages, you can see that the papacy ran alongside the state, sometimes sort of vying for power, but not against each other.
I always considered that battle of the milvian bridge, as that sort of turning point.
Is this interpretation not correct?