He literally loved her so much that he couldn’t bring himself to speak to her because it killed him inside! For a person who’s been faced with the choice to put down a pet or to continue/take off a loved one’s life support, that scene resonates to the point where I’m shaking at the accusation that he didn’t care and just wanted to get rid of her!
He knew that both of them were suffering and he didn’t know how to help them both. The longer he hung onto her, the more she hurt, and if he let her go, the more he hurt. It’s an impossible position to be in. Holding on is selfish and ignoring the situation is selfish—I happen to know that firsthand—but if you procrastinate the situation and trick yourself into focusing on other things, it gets you through another minute or another hour or day or week.
River was the one begging The Doctor for closure, he was the one who refused to let go of her because endings, especially hers, cut him too deeply. She was basically saying, “Please end my suffering; let me off life support,” and he was fighting that, because how do you say goodbye? And so she came up with a way for him to give her what she needed, without him having to do the thing he couldn’t. It was one of the most simultaneously majestic and tragic scenes I’ve ever seen!
And all too real.