I'm going off-track for a moment first - more than Satan not caring about his sons, I think Satan didn't expect that they would attempt to stop him from fulfilling Yuri's dream. Frankly, I kind of sympathized with him at the ending of the first season because he didn't understand why they would go against him even after his explanation.
"Why... Why would the brats born from me and Yuri destroy our dream?"
Satan is obsessed. He's obsessively trying to make Yuri's dream come true. Yuri is the only one who has shown him pure, altruistic love, and he's trying to honor her memory in his own way. He's practically on the brink of madness trying to make her dream happen - if his sons happen to stand in the way, he's ready to give them up because he cares a heck ton more about Yuri than them. Remember: they never had any interaction before their clash in the last few episodes.
Putting that aside, I think no, Satan does not understand Yuri's dream of peace. Like what Mindwin pointed out, Satan interpreted the idea of co-existence between the entities in a completely different way.
We have to remember that humans are the ones who almost killed Yuri - Satan was driven by desperation as he possessed one vessel after another in order to save Yuri.
"... Demons and humans can understand each other."
That's honestly what Yuri's dream is all about. But that's impossible if Satan is not able to exist in Assiah to understand humans (if he tries to).
There's a lack of understanding on Satan's part: from his perspective, humans are filled with animosity towards demons - true peace can only be achieved when there is only one ruler. His idea of merging the two worlds is to force the humans in Assiah to submit to him because he doesn't see any other feasible way to do it. In his attempt to achieve his own definition of peace, he thinks he is trying to fulfill what Yuri wished.
Obviously, if Yuri is still alive, this tragedy wouldn't have occurred. Because of all the love she has given Satan, she actually holds a significant amount of influence over his thoughts and decisions. If she knows that Satan intends for the violent, brutal approach, she'd have stopped him, in one way or another.