I think the problem with proposing counterfactual scenarios in which Constantine (and further Roman emperors, for the sake of argument) never convert to Christianity, is that religions undergo a process of memetic natural selection where Polytheistic beliefs are less resilient and have less survival value than Monotheistic ones, and like an inexorable tide, will eventually be drowned out. Polytheism like that of the ancient Greeks and Romans is too malleable to be taken as seriously by its adherants as any flavor of Judaism, Christianity, or Islam. The fact that new gods were regularly added to the pantheons of ancient Polytheistic beliefs, often mortals who attained a high office in life (Emperors, Kings, Pharoahs, etc), must communicate on some level to the believer that their religion was not a monolithic, immutable insight into the mysteries of the world, while the major Monotheistic faiths were able to lay a strong claim to the province of mystery for their believers. The 'closed' nature of Monotheism, and the extreme abstract and contradictory characteristics attached to the central unitary deity, help shore up the 'weaknesses' of Polytheistic beliefs from a marketing standpoint - it's harder for a simple mind to find the faults of Monotheistic belief than the alternative. It's easier for a believer to swallow the whole concept behind Christianity or Islam or Judaism or Zoroastrianism in one gulp, and therefore harder to dislodge, because once imbibed the central "truths" of Monotheism appear to have much more internal consistency and cohesion.

Christianity was gaining popularity in Rome during the periods of miltary anarchy that stretched the time span from the conclusion of the disastrous reign of Commodus in 192 AD (not the version shown in Gladiator, but the real-life Commodus was quite the nutjob and his policies quite destructive of the social fabric of Rome) and Constantine's Edict of Milan in 313 that removed the penalties associated with declaring oneself a Christian. The religion was seen as richer in insight and less vain than the evidently artificial, piecemeal Roman religion, appealing to the masses and women because it promised a sort of equalization between people of all social status in death. Constantine and the emperors following him surely recognized the vigorous, tenacious religion as a valuable mortar to cement a fracturing society, but in the end the Christian focus on the afterlife over the 'real world' quite probably contributed to the Empire's decline and caused the Romans to deal with the 'Barbarian' nations on their borders in a lax, detached way rather than aggressively assert the solidarity of their nation.

I think that, even if Constantine had never converted, or had lost the battle of Milvian bridge (the famous battle in which he attributed his success to the Christian god), the forces of time and public pressure would place another Christian emperor on the throne, who would commence similar reforms. Christianity was simply bound to outcompete the depleted, enervated Roman polytheism by its apprarent cohesion, stronger and more assertive truth claims, and its offer to the vast masses of humanity a share of divinity reserved only for kings and emperors in past faiths. What effect a later adoption of Christianity would have on the declining Roman empire is left open-ended, because the religion was at first a positive force to unite the citizens of the far-flung provinces, but in the end it allowed people to accept 'Barbarian' conquest and rule without vigorous resistence. Here's a scary thought: Perhaps a non-Christian Rome would stand tall against the Germanic hordes, only to be conquered from the South by energetic Muslim armies, paving the way for an Islamic empire stretching from the Nile to the Thames, and from Spain to the Danube? That's the danger of speculating counterfactual history: no one can possibly have a full grasp of all the variables, and the further from the present you travel, the more variables there are and the wilder history can diverge from its established course.

To be sure, ancient Greece was a hotbed of philosophical debate and malleable beliefs. I think that most Polytheistic religions are actually a lot closer to atheism than we suspect, as the transparent nature of the stories of gods and goddesses, the fact that they underwent literary evolution on a much greater scale than the easily-ossified Monotheistic faths, led people to treat the stories as useful allegory and metaphor rather than literal truth to be rigorously defended. I think the substance of what you're asking is, how would the ancients of a more open, philosophical time accommodate all the scientific discoveries pertaining to the workings of the world achieved to date (that is, the kind that destroy rigid doctrinal monotheism) with whatever religious traditions they maintained?

I think the ancient Greeks, who already embraced the open investigation of natural phenomena, would in the face of mounting evidence relegate their gods to a progressively marginalized role in their explanatory power, but continued to pass on their stories to confer the moral principles within. At least, that's an idealist's perspective. Since such a nation could not exist within a vacuum, the greater likelihood is that it would, in time, succumb to interal forces of decay and stagnation or external invaders seeking land, riches, refuge, or simply to spread their own, more aggressive religious beliefs. To suppose one democratic culture survive from the high Hellenic era is to suppose that none of the violent, expansionist cultures in history come into fatal contact with it - the Persian empire, the Roman republic (and later empire), the Huns, the later 'barbarian nations, the Christian crusader kingdoms & empires, the Islamic empire, the Mongols, Tamerlane, the Ottomans, the British, Dutch, French and Spanish trading empires, the German and Nazi empires, the Stalinist hegemony of the 20th century - all of these would certainly wipe out any trace of the open, democratic, rationalist society of the sort we're talking about and plunder the riches of a nation that spends its time in philosophical pondering. It is more useful for leaders to encourage in their populace a fervent, aggressive religion as a means of cementing national identity (while giving up a portion to imaginary characters), domestic tranquility, and assuring obedience to a singular sovereign.

Seneca the younger, the Roman philosopher, had a great quote on this subject -  "Religion is regarded by the common people as true, by the wise as false, and by rulers as useful." Rulers, and nations, best able to tap into the strength of a populace of zealots have historically always overrun nations engaged in more rational, open discourse. Of course that sounds incredibly pessimistic in these modern times, but of course our hope here is that humanity has finally reached some kind of critical mass where our interconnectedness, our sophistocated communication, and the triumph of reason-based initiatives in all fields of study (technology, travel, medicine, meteorology, agriculture, etc) will prevent us from sliding back towards conquest by hordes of religiously actuated 'barbarians'.