This nation's governors voiced apposition yesterday to two major elements of President Reagan's emerging domestic policy: his proposal to put a cap on Medicaid and his desire to shift responsibility for welfare back to the states.

In one of the largest budget cuts on his list last week, Reagan proposed a cap on the federal share of Medicaid, which almost surely would force states to cut back on health care for the poor next year. The governors told Reagan yesterday that they oppose this, think Medicaid should remain a largely federal responsibility, and think welfare should as well.

George Gov. George Busbee (D) also told the governors made it clear to the president that his proposed cuts in federal spending are "completely unacceptable" if not accompanied by changes in laws and regulations to let the states adjust the new budget levels.

The governors kept their criticisms muted for the most part. "We're concerned, but generally we're supportive," said Busbee, chairman of the National Governors Association. They met with the president and his top domestic policy advisers for about an hour in the East Room late yesterday afternoon.

Earlier, they had expressed their concerns a bit more sharply in meetings with administration officials.

At a session with Health and Human Services Secretary Richard Schweiker, Utah Gov. Scott Matheson (D) read a statement calling the Reagan proposal on Medicaid "not acceptable" and proposing an alternative that would, among other things, require the federal government to set its own interim limit on Medicare hospital reimbursement increases.

Matheson, the head of the governors association task force on health, said the statement represented the organization's policy views. Schweiker promised to consider it.

But Schweiker and Robert Carlson, a special assistant to Reagan, showed more resistance to suggestions from several governors that the president reconsider his intention to turn to the basic welfare program -- aid to families with dependent children -- back to the states in the form of a new block grant.

That welfare proposal was not included in Reagan's economic message to Congress last week, but the two administration officials left no doubt that Reagan would pursue the idea later.

They urged the governors' association to abandon its long-held position in favor of full federal financing and administration of the AFDC program, but were challenged on that point not only by Democratic governors but also by Tennessee Gov. Lamar Alexander (R), the governors' liaison with the White House.

Alexander and Busbee said they were encouraged that the administration is "not fixed in concrete on a lot of things," but declined to be specific. They made it clear, however, that the governors did not want a full-scale fight with the president and that discussions of welfare programs would come in "phase two" of their negotiations with the Reagan administration.

The more immediate concern of the governors was Reagan's proposal to put a 5 percent cap for fiscal 1982 on growth in the multibillion-dollar Medicaid program for the elderly and needy and to allow the program to grow no faster than the overall inflation rate in future years.

Reagan projects that this would save the federal budget $1 billion the first year, with the savings rising to $5 billion by 1986. His message to Congress promised legislative changes that would let states alter the eligibility, benefit and payment provisions of their programs "in order to meet the essential health care needs of their needy citizens" without "being forced to shoulder unreasonable additional burdens."

But Matheson warned that the 5 percent cap would take effect in a year when hospital costs are expected to rise by 18 percent. Delaware Gov. Pierre S. duPont (R), like Busbee, expressed concern that Congress might pass the cap without giving states the promised flexibility.

As an alternative, the governors suggested that the federal government put a 10 percent cap in 1982 on the Medicare program's reimbursement to hospitals, while giving the states much more flexibility in trimming their Medicaid costs and long-term health care programs.

Matheson said that approach would save the federal government $1.7 billion in fiscal 1982 on direct Medicare expenses and "make it far more feasible for Medicaid programs to impose similar restrictions."

Schweiker said he "needed to look at that alternative," but Congress, in the past, has refused to challenge the hospital lobby by attempting to put direct cost controls on hospital services.

The governors also expressed opposition to some Reagan tax proposals.

New York Gov. Hugh Carey (D) said the proposed 30 percent tax cut for individuals over three years "plain won't work." He predicted it would increase inflation.

Carey said he expects the Democratic-controlled House Ways and Means Committee to revise the president's tax program, and he urged that capital gains taxes, not personal income taxes, be cut. Govs. Richard Thornburgh (R-Pa.) and James Rhodes (R-Ohio) said the administration's tax proposals seemed geared to benefit new industries and not to help rehabilitate outmoded factories in the ailing industrial North.

Norman Ture, undersecretary of treasury for tax economic affairs, told the governors that he agreed some industries would get a better break than others, but that the administration's intent is simplify tax laws.

Rhodes responded sharply: "I think the most important thing is to get people back to work in steel, rubber and the automobile industry."

Reagan welcomed the governors to the White House and said he looked forward to working with them to realize "a longtime dream of mine" -- the return of many powers of the state and local governments. The president is establishing a task force on federalism to work on this procedure, and he asked for the governors' advice.