The fallout from the controversy also extended to Sinn Fein, with a prominent TD breaking ranks and refusing to support his party's call for abortion laws.
The 31-year-old died after she suffered a miscarriage and was refused an abortion at University College Hospital in Galway.
Sinn Fein rising star Peadar Toibin didn't sign a party motion calling for X Case legislation because he is pro-life.
The Government will have to give an interim response on their position in the Dail next week, as a result of that motion, but won't be discussing an abortion strategy until a week later.
Highlighting the gap between their stances, Mr Kenny said he wouldn't be "rushed" into a decision and Mr Gilmore said a decision "will not be delayed".
"We are going to take action on this. Doing nothing is not an option," the Tanaiste said.
The Cabinet is expected to make a firm decision on its response to a report by an expert group on abortion in 10 days' time.
Dr Reilly will bring the report to ministers at the Cabinet meeting on November 27.
Labour is demanding a recommendation from the minister and that a decision be taken that day – not just the publication of the report. But Dr Reilly was forced to promise last summer he would go back to the Fine Gael parliamentary party before a Cabinet decision.
Given the tight timeframe, this doesn't appear possible as Labour insist action will be taken once the report comes to Cabinet.
"We'd be expecting recommendations from the minister by then," a senior party source said. The Indian Ambassador to Ireland, Debashish Chakravarti, met Mr Gilmore at the Department of Foreign Affairs in Dublin.
Mr Chakravarti said the Indian government did not want to be involved into the inquiries into Savita's death, but asked that her family were.
"I'm under instructions from my government to approach the Irish authorities and to convey our concern, considering this huge human tragedy and to hear what the Irish side has to say," Mr Chakravarti said on his way into the meeting.
"We have said the family should be involved and it should be an independent inquiry."
Fine Gael TDs are firmly of the view that they will be consulted before a Government decision is taken and there will be a gap between the publication of the report and a Cabinet decision.
Fine Gael parliamentary party chairman Charlie Flanagan said it was his understanding the report would be discussed before the decision.
"The matter will be considered by the Fine Gael parliamentary party before a decision is taken," he said.
Mr Kenny said the recommendations and the report will be brought to Government by Dr Reilly and it will then be published.
After that, he said the Oireachtas "and everybody else" can discuss it and the Government has to send a report to a European body – the Committee of Ministers in the Council of Europe – by the end of November.
Mr Kenny denied the Government was at odds over the speed at which the legislation should be implemented, insisting "differences of opinion are only that".
"This is a matter that has divided Irish society now for a great number of years and I'm not going to be rushed into a situation by force of numbers on any side. This is something that has to be treated rationally and openly and truthfully and that's what will happen," he said.
But Mr Gilmore insisted the issue "will not be delayed".
"The terms of reference of the A, B, C group emphasize that this is an issue that will not be delayed," he said.
"This is something we will be dealing with and of course, we are required, in any event, to make a report to the Council of Europe by the end of the month.
"We have to examine the report, the Government has to discuss it. We have to hear what the Minister for Health has had to say about it."