In one of a series of messages on the social networking site on Monday, the 31-year-old all-rounder wrote: "I hope it was slow and painful".
The tweets were removed after Cricket Ireland issued a statement explaining that the player faces disciplinary action.
Mooney has now tweeted an apology to anyone who was upset by his comments.
He said: "I would like to apologise to anyone that I upset with my tweets yesterday regarding the death of Margaret Thatcher.
"I realise now that they were offensive to many and have deleted them. I'd like to assure my family, friends, employers and team mates that I have learned a very valuable lesson and in future will stick to focusing on my game!"
In its statement, Cricket Ireland said: "Cricket Ireland would like to distance itself from the comments made by John Mooney earlier today on the social network Twitter following the announcement of the death of former British Prime Minister Margaret Thatcher.
"Cricket Ireland have asked the player to remove the comments which have caused offence, and will consider appropriate disciplinary action in due course."
Former Prime Minister Margaret Thatcher died on Monday morning following a stroke.
She was 87 years old.