No, they couldn't do that. Here is how an owner who really gives a damn is likely to deal with a player who stages a protest during the national anthem:
1. Do nothing this season (for the reasons I documented above).
2. Cut the player in the off-season, as permitted under the collective bargaining agreement.
Item #2 as permitted under the CBA allows a team to get out from under a bad contract if the player's performance doesn't warrant the salary he is due to be paid for the remainder of his contract. It also allows the team to cut the player simply because the team wants to go in a new direction and the player's skills and style may not fit the needs of the new offense/defense/etc. the team wants to implement. Teams can also cut players like this if they have documented cases of player misconduct off the field.
My suspicion is that if the team gives any indication that they player is being cut for reasons totally unrelated to performance or documented cases of unacceptable off-the-field behavior, the player may have legitimate grounds to file a grievance under the CBA. That is why a team's management is likely to say absolutely NOTHING about the player's protest before cutting him.
“No, they couldn’t do that.”
Why not?
Players were threatened for kneeling on the field during a game, so there’s plenty of precedent for threatening them for kneeling on the field during a game - on grounds that both irritate/offend many in the audience.