The embryology bill has already prompted protests over abortion law
Anti-abortion MPs have criticised the decision to postpone debate on the controversial Human Fertilisation and Embryology Bill for three months.
The remaining stages were due to be debated next week but MPs were told there were scheduling difficulties.
Labour MP Jim Dobbin and Tory Nadine Dorries suggested with the upcoming Glasgow East by-election, the government wanted to avoid an "uproar".
Harriet Harman said time was restricted in the weeks before recess.
The controversial legislation - which covers stem cell research and abortion - was due to be discussed in the Commons on Monday but has now been delayed until the autumn.
Earlier debates had already seen MPs voting to reject a reduction in the time limit for abortions and to scrap laws forcing clinics to consider the "need for a father" before allowing women to seek IVF treatment.
The government, faced with the prospect of a rebellion by Roman Catholic ministers, allowed Labour MPs a free vote on the three most contentious issues during those stages.
But the Labour votes were to be whipped at the bill's third reading and there had been reports that Catholic Cabinet minister Ruth Kelly had been given permission not to attend next week's vote.
A new series of amendments had already been put forward ahead of Monday's debate in efforts to change abortion law.
But in the Commons at her regular Thursday session setting out the business for the week ahead, Commons Leader Ms Harman told MPs that the bill would not now be debated for at least three months.
Her Tory shadow, Theresa May, said Health Secretary Alan Johnson had described it as a "flagship government bill" and asked: "Can we have an explanation from the Rt Hon lady as to why this bill has been pulled from next week's business, at the last possible moment?
"And will she give an absolute guarantee to this House that this decision had nothing whatsoever to do with the forthcoming Glasgow East by-election?"
Later Nadine Dorries, the Conservative MP who has led calls to reduce the time limit for abortions, also said she believed it was postponed to avoid alienating large numbers of Catholic voters in the constituency.
She said: "Liberalising amendments on abortion that were being championed by its own MPs were deeply unpopular with the public at large, including those in Glasgow East."
And Mr Dobbin, chairman of the all-party parliamentary pro-life group, said the government was handling the legislation in an "insensitive" way, adding: "The likely reason is the enormous public uproar the bill, with its hugely controversial proposals, has had - particularly in areas such as Glasgow East."
But in the Commons, Ms Harman said it had been postponed to allow more time for debate.
She told MPs: "Of course the Bill remains a flagship government bill but in the last full week before the House rises difficult decisions have to be taken about what should be included by way of business."
For the Liberal Democrats, Simon Hughes said it looked "from the outside as if it is the government losing control of its business management".
He called for a promise that MPs would have two days to debate the bill in order to do "justice to the issues".
Ms Harman said she could not "guarantee" two days of debate but would try to ensure the discussion took place on a day that was not "carved out" by hour-long statements.
Asked later about the reasons for the delay later, the prime minister's spokesman said: "Harriet Harman gave a very full explanation of this matter in the House."
Liberal Democrat MP Evan Harris and former Labour Health Secretary Frank Dobson, who had put down amendments making changes to abortion law - such as reducing the need for two doctors' approval to one doctor - have welcomed the delay.
The Glasgow East by-election, which will be held on 24 July, was triggered by Labour MP David Marshall's resignation on health grounds.