Fermanagh and South Tyrone representative Phil Flanagan favourited and shared the Tweet with his 3,600-plus followers on Monday.
He apologised in a statement to the Impartial Reporter newspaper.
"In a split second error of judgement, I misconstrued the meaning of this tweet and re-tweeted it to my followers in that context," Mr Flanagan said.
I innocently believed that this joke was about an expectant father being spotted entering the hospital through the front door, leading to a two day media circus with no news to report
"In hindsight, I can now understand the potential double meaning that some may read into this, but this was not my intention.
"I apologise for any offence that this may have caused anyone."
Mr Flanagan was criticised on the social networking site after he shared the tweet.
It was posted ahead of the news that the Royal couple had given birth to their first child, who has been named George Alexander Louis.
He will be third in line to the throne and will be styled as the Prince of Cambridge.
TUV leader Jim Allister later confirmed that he had submitted a formal complaint about the tweet.
"While hardly surprising, it is nonetheless unwarranted and I would argue it was a breach of the Code of Conduct that all MLAs are bound by," the North Antrim MLA said.
"Therefore, I have today sent a letter of formal complaint to the Assembly Commissioner for investigation."