Senators Mike Lee and Jerry Moran each announced they could not support the bill in its current form, bringing to four the number of Republican senators opposed and effectively killing the revised version of the bill.
The delay is a huge blow to Trump and Republicans who have spent the last seven years plotting an end to the Affordable Care Act, commonly known as Obamacare.
A healthcare bill being unveiled by US Senate Republicans this week is expected to roll back the Obamacare expansion of the Medicaid healthcare programme for the poor and reshape subsidies to low-income people buying private insurance.
Ossoff and Handel both tried to focus on local issues and avoided mentioning Trump, whose approval rating sits at 37 percent, according to the latest polls.
US President Donald Trump and Australia's Prime Minister Malcolm Turnbull met for a patch-things-up summit in New York, with the US president saying an earlier telephone spat was "all worked out."
US Congress is scheduled to vote on a newly revised healthcare legislation on Thursday.
House Speaker Paul Ryan withdrew the legislation after Trump asked him to halt debate without a vote.
Opposition Democrats have warned against repealing the former president's healthcare plan, saying it will leave tens of millions of Americans without access to affordable medical care.
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