I kind of look at the scene like this, Han may have thought there was no saving Ben, saying something to Leia like "he has too much Vader in him." As close as Han and Chewie were, I think it's plausible that Chewie may have felt the same way or had at least come to the same conclusion over the years. It wasn't until Leia's assertion that Ben still had light in him that even Han thought maybe he could be saved. Once Kylo/Ben murdered Han, those initial feelings were all confirmed. It was time for Ben to be put down. And Chewie did what needed to be done regardless of Leia.
With this thinking, you can also possibly explain why Chewie and Leia don't hug. In his grief, he may irrationally blame her for encouraging Han to confront Ben and causing his death. He obviously realizes shortly after that isn't fair to Leia and lets that anger go. But, initially, he may have been cycling through a lot of different emotions.