The Health Minister Simon Harris has said it is "grotesque and disgusting" to think the details of a patient who had a medical abortion could have been leaked.
It was also alleged that the woman was later contacted by a bogus counselling service and verbally abused.
The woman had received a medical termination at Holles Street Maternity Hospital.
She then allegedly received a text message which directed her to an address on Dublin's Dorset Street.
An image of a text message she allegedly received has been circulating online.
The Health Service Executive (HSE) was asked by Minister Harris to look into the breach.
He said: "The idea that anybody might leak a woman's confidential information is reprehensible, it's grotesque, it's disgusting.
"And that's why I asked the HSE yesterday to investigate the matter and to report back".
"Worrying and sinister"
However Sinn Féin has said the issue should be referred to the Gardaí.
The party's health spokesperson is Louise O'Reilly: "I have written to the Minister for Health and the Minister for Justice on this matter and I am also calling on the Gardaí to investigate it as a matter of urgency.
"The crime which has been allegedly committed, a breach of confidential patient information, is a serious one in itself, but there was an alleged worrying and sinister attempt to use that data to lure a woman to an address on Dublin's Dorset Street for a fake medical appointment.
"The details of what happened, which have been circulated widely on social media, are very disturbing."
She added: "This is the second time in January that I have had to contact the Minister for Health in relation to the leaking of confidential details and it is very worrying and frustrating.
"This situation as reported is extremely disturbing and it leaves me concerned for the safety of all women in this State who may need to access abortion services."
Ms O'Reilly said while she welcomed the HSE investigation, it should be a criminal matter.
Additional reporting: Alison Nulty