Miroku and Sango had been outside with their twin girls, having just recently made the announcement they were expecting again. Shippou was playing with the girls while Kaede sat watching in the shade of a nearby tree. The scene morphed from watching his friends enjoying each other's company into watching a happy family, a scene filled with love and laughter. He knew these people considered him family as well, but this was not his family. Without Kagome InuYasha knew he would never have a family of his own. The pain of her loss, along with the loss of a family that never had the chance to exist, was more than InuYasha could bear, causing him to take to the trees.
When InuYasha ran, bounding from tree to tree, his spirit would settle somewhat. He never settled completely like he had when Kagome was with him, or to the emotional wasteland he was before he met her but it was better than when he sat and dwelled. Before Kagome, he never allowed himself to feel anything because it was a sign of weakness. If an enemy knew he had a weakness they would use it against him in battle. He never allowed anyone to know his life was painful for him. He never allowed anyone to know he had feelings about anything at all. After meeting Kagome, however, his entire persona was assaulted with a whirlwind of emotions he never cared to know about. Toward the end of their time together all these emotions had made him a better human and a stronger demon, and he discovered loved her. So much so he was willing to return to the feudal era without her. Now there was just grief for everything about her that was gone.
InuYasha tried so hard from day to day to be like he had been before her, feeling nothing, allowing nothing to bother him, guarding himself. It was exhausting! Besides that, he remembered how unhappy and angry he had been, keeping so much pent up inside. He couldn't bring himself to be like that again and he knew Kagome wouldn't want that for him.
Instead, InuYasha lived with the grief. It was the least he could do to honor Kagome's memory. She had brought so much to his life and helped him improve in so many ways it would be an insult to bury her and how he felt now that she was gone deep inside him, never to see the light of day again. If she knew he was trying to run away from something, even something as non-dangerous as his feelings, she would sit him until his back broke. The thought made him snicker.