Mooney, best known for hitting the winning runs in Ireland's sensational victory over England at the 2011 World Cup, has subsequently deleted the message, and followed up with an apology.
The chief executive of Cricket Ireland, Warren Deutrom, instructed the tweet to be removed, describing it as "crass, insensitive, and offensive".
"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!"
Mooney, who has represented Ireland since 2004, may subsequently face punishment, according to Cricket Ireland.
"Cricket Ireland have asked the player to remove the comments which have caused offence, and will consider appropriate disciplinary action in due course," a statement read.