On Thursday, the governor said he isn't sure the legislation will make it to his desk, and he still plans to postpone his decision on the program until sometime this summer.
Herbert said he is awaiting the results of an outside study on the expansion's effect on Utah.
There's no timeline for the study's completion and no federal deadline for Herbert to make a decision.
Proponents of expanding Medicaid say Anderegg's bill would tie the state's hands before the detailed analysis is available.
"This undermines the thoughtful deliberation and study that governor already paid for," said Judi Hilman, executive director of the Utah Health Policy Project. "This is not a decision that should be decided by politics or philosophy about the size of government."
Utah Democratic Party Executive Director Matt Lyon said the expansion will keep more tax dollars in Utah and alleviate some costs that the state is already paying.
"If we refuse the expansion, we will be on the hook for the hospital bills that the uninsured can't pay, and the money that Utahans paid into the system through our taxes will go to cover the Medicaid expansion in other states that did opt-in, like New Jersey and Arizona," he said in a statement issued Thursday.
Lyon called the legislation "a partisan hack job and ideological chest-thumping."
Lyon and other critics accuse Republicans of rushing the legislation through the process at the last minute. It appeared late Wednesday when lawmakers substituted it for a bill that originally sought to nullify the federal health care law in Utah.
Anderegg said he and fellow lawmakers altered the bill because a review by legislative attorneys stated the nullification would not likely hold up in court.
The House is expected to vote on the legislation early next week. Anderegg and his supporters hope it to have it approved by the Senate and head to the governor's desk before the legislative session ends Thursday.