"There is some urgency here because of what is happening in the individual market," the veteran Republican added, as GOP Senators say it's clear that Obamacare is not the answer for millions of Americans.
As for Democrats in the Senate - who are doing all they can to save the basic structure of the Affordable Care Act - they quickly made clear they strongly oppose the House bill and want to see bipartisan cooperation in the Senate.
"You've got to come meet in the middle and build bipartisan consensus," said Sen. Bill Nelson (D-FL), who argues the House bill threatens Medicaid coverage for seniors in his home state.
But when Republicans hear calls from Democrats for bipartisanship on health care, they remember that didn't exactly happen in 2009 and 2010, when the Obama health law was being cobbled together.
"As I said with Obamacare, anything that effects so many people and such a big part of the economy should have a solution that can attract bipartisanship," said Sen. Charles Grassley (R-IA).
"Obamacare didn't achieve that," Grassley said in a statement.
Not only are there differences between the parties, but also between the House and Senate, as several Republican Senators quickly made clear that they don't like details of the House GOP plan that was approved on Thursday.
"I continue to have concerns that this bill does not do enough to protect Ohio's Medicaid expansion population, especially those who are receiving treatment for heroin and prescription drug abuse," said Sen. Rob Portman (R-OH).
"Although I will carefully review the legislation the House passed today, there seem to be more questions than answers about its consequences," said Sen. Susan Collins (R-ME).
One obvious fault line for Republicans is that the margin for error is very small in the Senate, where the GOP holds only a 52-48 majority - and depending on the issue - there are multiple policy stress points.
Another difference is that under Senate rules, Senators can't even vote on a health care plan without the Congressional Budget Office weighing in on how much the plan would cost.
Not only will policy disputes cause trouble, but also the restrictive rules of the Senate under budget reconciliation, which could thwart some GOP changes to the Obama health law.
The House is out of session next week - but the Senate will be working, and facing a lot of questions about the road ahead on health care reform.
How swiftly that GOP work gets underway is not clear; it would probably be a big achievement to get a health care bill ready for Senate floor consideration by July.