Rios glared at Yankees starter Hiroki Kuroda after he got hit but calmly went to first base.
Rios was later replaced in right field by Daniel Robertson.
Rios is the subject of trade rumors and could be dealt by Thursday's non-waiver trade deadline. The Rangers have a team option on him for 2015 and are open to bringing him back. But if there's an opportunity to acquire quality prospects, the club wouldn't hesitate dealing him.
"I would like to stay here," Rios said before Wednesday's game. "At the same time you have to understand what the situation is and a few other things. It's a business and you have to look at it as that. We do enjoy what we do as players."