It depends on how well you know him and his daughter.
Just because he says he isn't going to kill himself doesn't make it that he is thinking of it...we all think of severe depression and suicides...
So, maybe he is thinking you think that way.
If you are so concerned, call his daughter...explain what he has been going through...and what his response is to your 'concern'.

I will add this...sometimes a persons 'concern' feels intrusive...in physical illness and especially with depression.
If he is adamant about not going into treatment, the only thing possible is to contact his regular doctor and report to him your concerns.
Since he continues to ask you not to be 'there for him' so much, the best thing to do is respect his wishes...why?
Because a lot of depression has to do with boundaries...and making them, then having someone else, who knows better of course, tell us we can't have that boundary...too bad.
So, we build walls, higher and higher...until no one can breach them...nor can we breach them ourselves...we are locked into a prison of our own making.