Mr Poots said he had accepted an apology from Mr Beattie, but said his judgment in posting the message was "poor”.
Mr Beattie deleted and apologised for the Saturday evening post on Twitter, which included an imagined scene where he and Mr Poots are in a barber shop and with a punchline that makes reference to Glynis Poots and a brothel.
Today Mr Beattie repeated his apology, saying he was “genuinely sorry”.
In the Assembly, he said his language was “wrong at every level”.
But speaking to the Nolan Show, Mr Poots said it was wrong for anyone to draw family members into political debate, saying they are entitled to privacy.
"I just don't know what he was thinking about," he told Nolan.
"I'm a big boy, I can take a lot of abuse in politics and that's fine, but [he] shouldn't be drawing other members of people's families into anything, they have their right to their private life and, unfortunately, that has been infringed when it comes to my family.”
Mr Poots said his wife "doesn't want to be involved in public life".
He added that the ‘joke’ “demeans women in general, and it does, but it demeans one woman in particular and that one woman happens to be my wife and my children's mother.”
Mr Beattie told BBC Radio Ulster's The Nolan Show: "At times we get things wrong and it is important to stand up and say we got it wrong and apologise and I have apologised.
"I posted it as a joke, sometimes we all have to mind our language, we have to be on guard of our language, I let my guard down and I have let people down and for that I am genuinely sorry.
"I have written to Edwin to apologise, I did so yesterday."
Asked if he had apologised to Mr Poots' wife, Mr Beattie said that if Mr Poots thought it "appropriate that I apologise in person then I certainly will apologise in person".
DUP deputy leader Paula Bradley condemned Mr Beattie’s post.
“This is an appalling comment to make about anyone,” she said.
“To seek to score a political point against an opponent in this manner is not befitting a leader. For the comments to be about a woman not even in public life is a further indication of Doug Beattie’s default setting.
“This was an offence to all women at a time when parties are working to stop attacks on women, particularly on social media. It’s no laughing matter.”
Simon Hoare MP, chair of the House of Commons NI Affairs Committee, retweeted Mr Beattie’s post, it emerged last night.
He deleted it after about 20 minutes.
He did not immediately respond to a request for comment.
The Ulster Unionist leader drew fire on social media.
Justice Minister Naomi Long said she was “genuinely shocked at just how casually misogynistic” the tweet was.
“It would have made many a 1970s comic blush,” the Alliance Party leader added.
After deleting the post and joke, which he said he did not come up with himself, Mr Beattie initially wrote: “Ok so my joke wasn’t to universal approval so as not to offend I deleted it.”
Early on Sunday morning he posted again: “Last night I posted a joke — I introduced it as a joke. It was an attempt at humour nothing more.
“In no way would I intentionally offend anyone but it’s clear I have so it is only right I apologise for doing so. We can all make mistakes, my choice of joke was a mistake.”
He included a short video of a child looking extremely glum.