Winning him over could be as simple as just having a talk with him.
If all the both of you do is stare and look away from time to time, nothing will be accomplished.
That argument may be something that can be easily resolved if the both of you avoid allowing emotion to get in the way and merely look at it from more of an objective point of view.
The both of you were getting along before this argument happened.
Hence, there is a good chance that the reason why the both of you are no longer communicating is due to that argument.
If you have a talk and get whatever issues that caused the argument resolved, that may open up the possibility of engaging with each other in more of a romantic capacity from that point on.