I'm so sorry I missed the comment you posted, um, last November. (Hides head in shame...).
Denethor likes forward, strong-willed women, having been brought up in a clan of them, so Eowyn would be right up his alley. He'd always maintain that Faramir was not quite good enough for her (though not really - he'd be proud of the kid for winning over such a cool wife). And the grand-babies would be spoiled like no one's business.
I don't think he could stay in Minas Tirith, especially after Arwen showed up, but he would do interesting things elsewhere. Much to Thorongil's annoyance. Kind of a permanent thorn in the side of the new king. The dynamic between them would be interesting, to say the least, with Denethor pulled between seething resentment and helpless devotion, while Aragorn tries to win his "brother" back and gain Denethor's trust, despite the opinion of pretty much everyone that the former Steward's hatred is implacable.
It all hinges on forgiveness, as the mariner said.
Thanks for reading,