As tensions mount, Taoiseach Enda Kenny insisted that the forthcoming changes would not lead to abortion being used "as a form of contraception".
But pro-life and pro-choice supporters are furiously lobbying in advance of the legislation, with some resorting to vicious letters and emails to put pressure on TDs.
Mr Ring, the junior minister for sport, told the Irish Independent that he had never got such abuse from both sides in the abortion debate in his entire political career.
He referred to an email he received this week from a named individual, although he doubted that the name was real.
"I was accused of murdering my future grandchildren," he said. "I hope we can have a rational debate on this. These kinds of threats are not acceptable."
The Mayo TD was also sent a letter recently from a man in Wexford asking him where he was going to bury the foetuses who died as a result of abortions. Another letter contained a photo of a dead foetus, which he described as the worst he had ever seen.
"The minute I got it, I put it straight in the bin in case any of my staff would see it. That's the kind of stuff we are getting," he said.
Mr Ring's comments came after European Affairs Minister Lucinda Creighton revealed that she has received vicious personal and threatening correspondence from both pro-life and pro-choice groups.
She also appealed for a rational debate, which was respectful of people's opinions on abortion.
It is the latest example of personalised abuse being directed at politicians on the issue.
One cabinet minister has opted to bin all anonymous letters on abortion and delete anonymous emails without reading them.
The Irish Independent contacted the offices of Mr Kenny, who has already described recently how he had received a threat in relation to his criticisms of the Vatican after the Cloyne Report.
A spokeswoman declined to go into detail about the nature of the abortion-related letters and emails sent to him.
A number of other ministers said they had received a large amount of correspondence on the divisive issue, but none said they had received "vicious" letters as Ms Creighton had.
A spokesman for Jobs Minister Richard Bruton said he had gotten a significant volume of abortion-related correspondence, which was almost all standardised lobbying by various groups.
"There has been nothing personalised," he said.
Education Minister Ruairi Quinn has received a large level of correspondence relating to legislating for the 1992 Supreme Court X Case, which allowed for abortion in the case of a pregnant teenage rape victim.
His spokesman said the vast majority of the correspondence was in favour of legislating for the X Case. "None of the correspondence to date has been untoward," he said.
The Oireachtas Health Committee is due to begin its hearings next week in the Seanad chamber on the Government's plan to introduce legislation plus regulation to allow for abortion where a mother's life is at risk.
Suicide will be included as a ground for abortion, although there will be strict procedures.
One political source said that the abortion-related correspondence sent to politicians usually depended on what constituency they represented and their position on the issue.
He said that Mr Kenny was an obvious target given his position as Taoiseach.
And he said Ms Creighton may have become a target because she was a young woman politician who was opposed to abortion.
Justice Minister Alan Shatter, who has spoken strongly about the need for the forthcoming legislation, has received no threatening letters.
But he recently had to condemn the placement of "obscene" anti-abortion posters in his Dublin South constituency containing bloodied photographs and the slogan "Every Child Matters Except to Alan Shatter".