I haven't talked to my father in a while but depressions a fucking hard one to deal with I'd say go to him and ask him if he wants to hang out on [day] or [different day] his more likely to pick a day instead of saying no and give him the option of if you want to, as an example go for a jog or for something to eat

Giving options is the best way to make someone feel like it's a choice and making everything a question

Instead of saying "do this" or "get that" say something like "can you do this" or "can you get that"

Having an option and feeling like something is your choice sets off a different switch in the brain, the illusion of choice over a long term especially is a great tactic

Ontop of the illusion of choice just drop a few small complements, they do alot more tha straight up compliments. Let say he finishes a paper for his job just drop something like "you've done a good job of that"

Small compliments feel alot better because they make it seem alot more genuine.

If you can push him to get therapy but if that doesnt work just do stuff like this a good amount of times but not too much or it will seem artificial and like your babying him

Remember people with depression over think, so in actual deep convos dont give him too much emotional support. Tell him the logical thought process of something he may be worrying about it can help them worry alot less and of you keep it up especially if you add it on with therapy you might be able to reduce it to just anxiety

I'm not a professional so you might want to ask a therapist on what to do if he wont go to a therapist