Mr Beattie said the joke was not his and he was sharing something that had been sent to him. He has now deleted the post.
The joke described a fictional scenario involving Upper Bann MLA Mr Beattie and Mr Poots in a barbershop. It made a reference to Mr Poots’ wife and a brothel.
“Last night I posted a joke – I introduced it as a joke. It was an attempt at humour nothing more,” Mr Beattie tweeted on Sunday.
“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.”
Mr Poots’ party colleague and DUP deputy leader Paula Bradley condemned the tweet.
“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.
“He has rightly removed the comment but his attempt at an apology is feeble and only a reaction.
“If he truly felt remorseful and understood the offence he caused to Mrs Poots, her children, Edwin and the wider family, he would have apologised to them directly.”
Alliance Party leader Naomi Long also criticised the remarks.
“Genuinely shocked at just how casually misogynistic that was,” she tweeted.
“It would have made many a 1970s comic blush.”
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