Ah, well then you know exactly what partner-bereavement is all about sadly....
You mention his GP - although the GP can't discuss him with you, there is nothing to stop you writing to his GP to tell him your worries, so that that can be 'input' for future reference. I don't think the GP has to say to him that you've written.
It could be, though, that there is nothing you can do now to 'cheer your father up'. He's gone into mourning, so to speak, for his wife, for the life that he used to have that is no more (we widows know all about that alas...), and is, in a way, mentally shutting down towards the end of his own life maybe??
Yes, he probably is depressed, and who would not be - but it could be simply 'sadness', and nothing therefore could 'cheer him up' other than his wife getting better, which she can't do now.
Any 'cure' has to come from within - again, as we widows know. He has to find in himself a reason to enjoy his life even in the limited way that is now possible. If he has neither the wish to do so, nor the ability to do so, then awful though this is, that is 'his choice'. We cannot force others to be happy (alas, this forum is full of members who have had to come to that conclusion).
Maybe all you can do is to speak robustly to him and say that just as YOU had to find a reason to get up each morning after being widowed, so must he - IF he wants to have any enjoyment of his life at all any more.
That said, counselling may help him reach that final stage of grief - 'acceptance' (which does not mean 'agreement'), which we know we have to reach in order to keep going.
How long has your mum been in residential care for her dementia (which does sound of a distressing kind, if it can be verbally aggressive)? That may give a clue as to 'how long' his 'mourning' can be expected to last.
One final point for now - please do do do be very very very wary of being 'sucked in' to his situation. When you said he complains of not seeing anyone, when you haven't been to visit, there is a red flag running up the flag pole, because I fear that what he is meaning 'I haven't seen YOU'......
DO be firm on this. He can't turn you into a substitute 'wife' for him - ie, someone to give him the constant companionship that a wife does. I've known this in my own case - my son is now a young adult and on NO NO NO NO account must I try and make his company 'companionable' in the way my husband's was! I think there can be a very real danger that a bereaved person flails about emotionally to try and find a 'substitute' in a way (I have to be careful of that in respect of my brother as well.)
So, you cannot become the person he becomes totally emotionally fixated on, dependent on - especially with your sister refusing to get involved (maybe she has a sense of self-preservation???????).
Like it or not, your father has now to get used to a great deal of his own company, and that, sadly, is that. It's up to him whether he choses to socially interact with others. But he can't 'collapse' on to you. Sad but true.