Mr Beattie has said he is “ashamed” by the historic tweets that have emerged since he apologised for joking about a brothel and a rival’s wife.
The tweets have seen Mr Beattie again accused of misogynistic behaviour.
Mr Poots said the tweet by the UUP leader on Saturday night was “incredibly hurtful”.
UUP leader Mr Beattie, apologised and reiterated that apology on the floor on the Assembly on Monday, insisting he was “truly sorry”.
However, hours later, he apologised again after other historic tweets started to circulate.
In a statement, Mr Beattie said: “Since I made my unreserved apology in relation to a tweet at the weekend I have been made aware of a number of historic tweets which I published.
“I am deeply ashamed and embarrassed by the content of these tweets. They are totally unacceptable. They do not reflect the opinions or values I hold today.
“From the time I entered politics I have made a serious effort to increase the representation of women and advocate policies which would improve the lives and wellbeing of all women and girls.
“That is where I stand and I deeply regret the fact that these historic tweets could serve to undermine that position.
Speaking to BBC Radio Ulster on Tuesday morning, he suggested he could quit his role as party leader if the UUP believe he should do so.
“If I go back 10 yeras, then there is a clear failing in myself and I have to own up to that,” he said when asked on the Radio Ulster’s Nolan Show if he is fighting for his political future.
“Therefore I have to look and see if this is the sort of person who has to lead the Ulster Unionist Party. I will speak to my MLA group today, and I will speak to my party officers through my chairman Danny Kennedy, and if either group feels I should step down then I will.”
He added: “Likewise if they think I should refer myself to the party executive or the wider council on a vote of no-confidence then I shall do that as well, and the party will decide whether or not they can continue to follow my leadership.”