Before going further I just want to note that Constantine I didn't necessarily "embrace" Christianity as the "right" religion, the Edict of Milan (313AD) promoted religious tolerance and freedom across the Empire, and did not establish it as the state religion. That was done by Emperors Gratian and Theodosius I via the Edict of Thessalonica in 380AD. While yes, Constantine was a major patron of the church and did assist in it's dramatic spread across the Empire, the argument could be made that later rulers had more influence upon its spread.

While Christians were a minority in the Roman Empire, and were heavily persecuted in the centuries preceding Constantine's rule, it had been spreading quite far during the period. One of the most important factors in this was the preaching, and establishing of churches, by the Jew Saul (biblical Paul) of Tarsus.

Paul was born a Roman citizen, due to his father, which gave him freedom to travel throughout the Empire unhindered. In brief, Paul originally partook in persecuting Christians, until eventually converting and becoming one of Christianity's largest supporters. He was responsible for many of the concepts which led to the rapid spread of Christianity, most notably the affirmation that one did not have to be a Jew to become a Christian. He traveled throughout the Mediterranean preaching, until finally running afoul of authorities, which eventually led to his execution.

Keep in mind, that during this time Christianity was seen as a heresy by the majority of the Jewish community, and began to draw the ire of authorities who saw it as a threat to their power. Also, the relatively secrecy in which practitioners had to operate (Roman Paganism still being law, Judaism being a special case), and some of the ambiguous terms used (the body and blood of Christ) led some to believe it may also be a cannibalistic cult. But even through all of this, Christianity was still continuing to spread.

In 64 CE a major portion of Rome was burned, and a number of people placed the blame on the now infamous Emperor Nero. While there has been argument over his involvement, this is the incident where he purportedly "Fiddled while Rome burned", he still had the need to redirect public anger and did so by blaming the Christians for initiating the fire. This resulted in mass persecution of Christians in Rome, in some of the most heinous ways used at the time.

Following this period, persecution was usually limited in scope and at the discretion of local authorities. Evidence of this can be seen in the letters of Pliny the Younger (Pliny Epistulae Vol X.96 and X.98 respectively) in which he asks Emperor Trajan for guidance in their persecution. In response, Trajan directs that he should not actively seek them out, but if they are identified and refuse to denounce their religion than he should execute them.

Under Emperors Diacletian and Galerius, Christianity saw mass persecutions once again become normal. This, in part, was due to interpretation of advice from the Oracle; though it should be noted that Galerius had been adamantly opposed to Christianity all along. Also worth noting that Constantius served as Diacletian's co-emperor at the time of this; and that Constantine I's mother was a practicing Christian. Even with it being official edict, and the widespread persecution, Christians continued to escape authorities and build in strength. Arguably, at this point, it was too heavily established to be totally destroyed.

I think, even without Constantine's direct intervention, we would still see Christianity today, though in a much more fractured state. We may have never seen the rivalry between East and West come to fruition, and most definitely would have seen the Roman Empire take a different route in its decline as a European power, and even an entirely different history in and around the Mediterranean and middle east. But that, I think, would be a better discussion for r/historicalwhatif.

This question cant really be answered because what you are suggesting did not happen and all answers would be speculation. Christianity does favor the poor and suggest equality so therefore it would have still likely spread or would have been overcome by something else (perhaps Islam).

Sorry, but your submission has been removed because we don't allow hypothetical questions. If possible, please feel free to rephrase the question so that it does not call for such speculation, and resubmit Otherwise, this sort of thing is better suited for /r/historicalwhatif.