I guess we will never know. It’s a very thin gold band, so it must be a ladies band. And, like you, I care because he’s my favorite actor. I’min my seventies and have always had a crush on him.

I'd say signet ring. And then I'd say who the heck cares.

Right, I'd have to agree with joystar on this because there is no evidence to believe that Cary and Randolph went with each other in the 1930's.da0 I mean, it would be okay if they did, but they allowed too many starlets to chase after them, and then they fought over Irene in front of Gail on screen.
Randolph was more of the fashionplate trendsetter, like Cary. I'd have to agree that it were a School or Military ring even though I didn't see much of that in his films.
And if joy's Grandfather wore one, then that cinches the matter because he had grandchildren, right?
_ ||_ () ooO(_)Ooo

It might also have been a School or Military ring. Also, many "Old School" men wore a signet ring as a wedding ring on the right pinky. My Grandfather did.
"I do hope he won't upset Henry.."

I'm sure if they were, they weren't talking about it.

Most gangsters wore pinky rings.I guess they were gay too???

I read on Cary Grant's page that there were rumors because these two lived together on and off.
From Wikipedia
The pinky ring was a fashionable jewelry accessory for male homosexuals during the 1950s through the 1970s.
You said from the 30s on but maybe he was a trend setter?