Nope. The chairman can only be removed by a 2/3 vote of the RNC. Now in most years, the nominee would have a lot of support from the members of the RNC, so they would go along with his desires. But Trump has made sure that he has almost no support from the RNC, so they have absolutely no reason to go along with him on this.
Good point. I should have used the word "ability" instead of "right".