Nope. UT cannot open itself up to a lawsuit by making a rash decision. They just can't. So, what did they do?
1.) They instantly suspended Beard, without pay, without access to the team.
2.) They seemed to form some sort of group to investigate this stuff on the side, on their own. This involved their own legal team and the police, I'd imagine.
3.) After spending a few (fairly short, over-holiday) weeks gathering info, they quickly made the decision to let Beard go based on that data.
4.) They apparently gave Beard the option to go peacefully, which he declined. They then followed that up with firing him for cause and then publicly mocking him for it after.
This all adds up to me. They handled it perfectly from a legal liability standpoint, and even retained to viewpoint of a moral high ground for firing him at all.
What possible lawsuit? They had the right to fire him immediately. I agree that they did the right thing by suspending him and then investigating, it's just the timeline of the investigation went on waaay too long. It should have been a couple days, not weeks. There could have been a lawsuit about if the firing was "with cause" or without, but that's still in play regardless. It's up to Beard and his lawyers to decide if they want to go after that.