Littering for one. I suspect a disorderly conduct type of charge could be made to stick for another.

Did the cop have the authority to make her apologize? No.
Do cops do things like that all the time? Yes.

Having had some conversations with cops about how they view enforcement, my understanding is this: If they see you kill someone, or steal something you are going to be arrested. But as for really minor things like jaywalking they view these laws as tools that they don't go out of their way to enforce, but will use if there is something else going on (e.g. a rowdy group on a pub crawl where the people involved are causing a disturbance might get ticketed to provide a disincentive for being on the street). Similarly a lot of times if a cop sees someone doing something stupid or dangerous, rather than dealing with the paperwork, they will use the threat of arrest in order to get them to stop doing whatever they are doing and if possible fix whatever they did.

I think that this is what happened in this case. The cop sees the woman pour the water on the floor and (quite reasonably IMHO) decided that the woman was being stupid and threatening the safety of others in her actions. The cop then aggressively goes to the woman and says (more or less) "Hey, that was a stupid thing to do. You are going to clean that up or you are going to jail." At one point the woman seems to just want to walk away, and the cop grabs her shoulder and tells her that she seriously needs to clean up the water or she will go to jail. At this point the cop is not going to let the woman defy her authority. The cop is really annoyed now and makes her apologize to the TSO before she can leave.

So while the cop has no authority to order her to apologize to the TSO, she did have the threat of arrest to make it in the woman's best interests to comply. Probably in most cases the humiliation would cause the person not to do something stupid like poor water on a polished stone floor of heavily traveled hallways (and likely that was a large part of the cops motivation). In this particular case it did not work out that way.