Is it unconstitutional to ban someone based solely on a previous conviction?

While it may not be ILLEGAL for a social media site to ban a person based ENTIRELY on the person's political views it does change the the social media's status from CLAIMING to be an open platform to actually being a politically motivated or driven platform that they're NOT advertising or disclosing to they're users. Now if they are being politically motivated without disclosing this fact to they're users while CLAIMING to be politically NEUTRAL THAT could be enough to cause them legal issues! Personally I would NOT register use ANY forum or service CLAIMING to be politically neutral while they're attacking they're users for simply having a differing political views WITHOUT the forum or service publically disclosing that information! To me this is NO different than a MSM outlet CLAIMING to be politically neutral while the information they CHOOSE to present and how they present it CLEARLY demonstrates political bias like our current LIEberal left socialist demoCROOK CONTROLLED alphabet MSM does DAILY!!!!

In case you haven't noticed the government has ceded a whole lot of power to social media to dictate whose voice can and cannot be heard on its platforms. This is a fight you can't win under current management. 

There is nothing in the constitution guaranteeing people to be free of discrimination, thus what may "feel" like discrimination or actually is in fact discrimination does not violate the US constitution.  US law does protect people from discrimination if they are members of protected classes and the discrimination is systematic.  Convicted criminals are not a protected class.  Thus the same right an employer has to not hire someone convicted of a crime, allows a social media platform the right to deny service.

The Constitution does not apply to social media, it is between you and the government. 

Discrimination is when you are treated different based on your race, nationality, or some other protected class. Being convicted is not a protected class. 

Not unconstitutional.  It is discrimination.  Legal discrimination.  Most discrimination IS legal.

You answered your own question: a social media website is a private company and they can make whatever rules they want. It's just like a store that bans you from entering the premises for shoplifting. Their property, their rules. It's only discrimination if you exclude a group based on race, gender, religion, etc.