Ross, who returned to his Radio 2 show on Saturday following a three month suspension for the "Sachsgate" affair, was just eight minutes and 35 seconds into the show before unleashing the remark.
He had been chatting on-air with his producer Andy Davies about how they spent their time during the suspension.
Davies said he had been in Spain bricklaying in the garden of his villa and kept getting "grabbed" by an elderly woman who was trying to kiss him.
Davies said: "She must be about 80 I reckon."
Ross said: "Eighty, oh God. I think you should, just for charity. Give her one last night, will you? One last night before the grave. Would it kill you?"
The conversation then ended abruptly before a song was played. The banter did not continue following the song.
The Daily Mail tracked down the lady's son, Jose Maria Moreno, who said: "It is offensive. My mother's mental health should not be a subject for comedy and Ross should be sacked.
"What he said is unforgivable and offensive. I don't understand how he can continue working for an organisation like the BBC."
Tory MP David Davies was reportedly listening to Ross's morning show on Saturday and also demanded his resignation.
He told the News of the World: "I was listening with my kids to this. There's a place for humour but it has to be appropriate to the time of the day. And that clearly wasn't.
"He should have gone ages ago. There's no way this man should be on the air. He needs to be replaced now."
Ross was suspended without pay on October 30 last year after thousands complained about a series of obscene phone messages that he and comedian Russell Brand left for 'Fawlty Towers' actor Andrew Sachs while broadcasting Brand's former Radio 2 show.
In one of the messages, Ross told Sachs that Brand "fucked" his granddaughter.
Brand resigned from his Radio 2 show shortly after the incident and was later followed by Radio 2 controller Lesley Douglas and David Barber, the station's head of specialist music and compliance.
Ross apologised for the incident and repeated his apology when his Friday night chat show, 'Friday Night with Jonathan Ross', returned last week.
Full Transcript of the Radio 2 incident on Saturday:
Davies: "There's a women in the village who's getting on a little bit, keeps trying to kiss me. Older woman. Very older woman. So I'm trying to do my brick work and she keeps coming up. Chatting away to me.
"Can't understand a word she says. And she keeps grabbing me. Every time I see her now I have to run indoors."
Ross: "She actually grabs you."
Davies: "Hmmm."
Ross: "And how passionate is the kiss that she attempts?"
Davies: "She's apparently got a thing for younger men."
Ross: "Good heavens above. It must be slim picking in that village."
Davis: "So I'm there with my trowel in my hand."
Ross: "Well that's the thing right there."
Davies: "That's what it is. I haven't got a belt."
Ross: "You can picture it now ladies and gentlemen. Him sitting out there stripped to the waist, hey. Stripped to the waist. Trowel in one hand, hey, lump of wet cement in the other. Looking magnificent. Sun glinting off what was once a proud chest and is now glooping over somewhat. And an old lady comes over."
Davies: "She must be about 80 I reckon."
Ross: "Eighty, oh God. I think you should, just for charity. Give her one last night, will you? One last night before the grave. Would it kill you?"
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