The announcement came after McCain's office said he would remain in Arizona next week following a procedure to remove a 2-inch (5-cm) blood clot from above his left eye.
The Mayo Clinic Hospital in Phoenix released a statement Saturday that said McCain's surgery Friday was successful: "The Senator is resting comfortably at home and is in good condition".
But this week will give senators extra time on nominations and to consider government spending bills and a debt ceiling increase, the deadline for which is now October.
Senate Minority Leader Chuck Schumer said Sunday that he did not think delaying the vote on it during McCain's recovery "changes the outcome".
The other Republican senator from Arizona praised McCain.
Until McCain's surgery, McConnell had hoped to vote on the Republican plan in the coming days.
McCain has said he's concerned about the impact of proposed Medicaid cuts on his home state, and said on July 13 he couldn't say whether he would support McConnell's new bill. "His Mayo Clinic doctors report that the surgery went "very well" and he is in good spirits", the Mayo Clinic said.
McCain's absence takes away a crucial vote for a Republican plan that's on a razor's edge for support.
McConnell and other party leaders have been urging senators to at least vote in favour of opening debate, which would allow senators to offer amendments. Cruz's additions to the bill has garnered widespread conservative support in the Senate, but has left many moderates cold. Medicaid- a social health care program for families and individuals have limited resources.
The question now is whether the Cruz amendment will now be scored by the CBO, given that a vote on the bill has been delayed.
"Arizona has been nationally recognized for running one of the most efficient and cost-effective Medicaid programs in the country", McCain said in a statement.
In his appearance on "This Week", Price countered by arguing that the Trump administration would be taking further administrative actions on health care and that the Senate health care bill is "not the entire plan".
Heather Carter, a Republican from Cave Creek, said cutting Medicaid threatens 400,000 people who have gained insurance through the Affordable Care Act's coverage expansion and also the state's health care system as a whole.