Taking responsibility for "a British humiliation" in last week's seizure of the Falkland Islands by Argentina, Britain's foreign secretary, Lord Carrington, resigned today as the armada seeking to restore control of the islands took off after a noisy send-off from about 5,000 flag-waving Britons.

Prime Minister Margaret Thatcher said she regretted Carrington's resignation but was unable to dissuade him because he made it "a point of honor."

Thatcher and Carrington sought to emphasize today that his resignation did not reflect any disagreement between them over British policy in the Falklands crisis. "Absolutely not," Carrington said, when asked that question tonight. "I agree 100 percent with what the prime minister has done, and I back her up to the hilt."

Thatcher quickly replaced Carrington with Francis Pym, 60, a former defense secretary and leader of the House of Commons.

As the vanguard of a powerful British fleet of 36 warships set sail this morning from Portsmouth on the southern coast of England for the South Atlantic, Thatcher also declared her embattled government's "quiet, professional determination to retake the Falklands because they are still British territory."

Defense Secretary John Nott, the operational chief of this military effort, also offered his resignation, but Thatcher refused to accept it. He had been criticized sharply along with Carrington in Parliament and the press for failing to avert the Argentine attack on the Falklands, but Thatcher said primary responsibility rested with the Foreign Office.

"It is vital you continue as our forces prepare for the possibility of armed action," Thatcher told Nott. "I have the fullest confidence in your ability to carry out the crucial tasks ahead."

Rex Hunt, the deposed British colonial governor of the Falklands, arrived here today and told of "very serious" fighting that took place when greatly outnumbered British Marines attempted to repulse the Argentine invasion. Hunt declared, "I intend to return. I am still governor of the Falkland Islands."

Resigning along with Carrington today were his deputy, Humphrey Atkins, a former Northern Ireland secretary, who admitted Saturday that he had misinformed Parliament about details of the government's knowledge of the invasion, and Richard Luce, the minister who tried unsuccessfully to negotiate a settlement of the dispute with Argentina.

Diplomats said the Thatcher government still appears uncertain about how it ultimately intends to use the fleet of aircraft carriers, submarines, destroyers, frigates and support ships being sent to the South Atlantic. But its talk continued to grow tougher today.

Asked on national television whether she also felt responsible for the loss of the Falklands, Thatcher said, "In the end, of course, the prime minister always does. That's why I feel deeply and thoroughly we must regain the Falkland Islands. They are still British."

When questioned about what would happen if she failed, she said, "I am not talking about failure. I am talking about my supreme confidence in the British fleet. . . .

"Failure?" she asked rhetorically. "Do you remember what Queen Victoria once said, 'Failure--the possibilities do not exist.' That is the way we look at it . . . We must go out calmly, quietly, to succeed."

Calling Carrington an "absolutely outstanding" foreign secretary, Thatcher said she "spent a lot of time Saturday and Sunday trying to persuade him not to put in his resignation." But when he "put this to me as a point of honor and said there had to be honor in politics," she said, "I was not at liberty to refuse."

Carrington said he thought the Foreign Office had done nothing seriously wrong in its handling of the long-running dispute with Argentina over the Falklands, "but obviously the result was wrong and there has been a British humiliation. I think I ought to take responsibility for it."

He acknowledged that he may have been mistaken in his assessment of Argentina's intentions, and "it was quite clear from newspapers and debate in the House of Commons Saturday that my judgment has been questioned."

In such a position, Carrington said, "you take the decision to go and you resign honorably."

Thatcher and Carrington often have disagreed on policy questions, sometimes loudly in the presence of others. But senior Conservatives and other knowledgeable observers have said that Thatcher respected Carrington for standing up to her, as well as for his confident approach to power politics and diplomacy. He is credited with moderating her views on many issues, especially how to handle the problem of settling the independence of Zimbabwe, formerly Rhodesia.

Politicians and analysts here expressed uncertainty about whether the loss of Carrington, who previously had been judged the most important and successful figure in Thatcher's Cabinet, would affect her efforts to resolve the increasingly threatening Falklands crisis.

They said his resignation could ease political pressure on the government, which will be tested by another emergency parliamentary debate on Wednesday. Carrington's influence in world affairs and on Thatcher, they suggested, could be offset by Pym's freedom from blame for the Falklands fiasco and his strong base in Parliament, possibly giving him more leeway to search for a peaceful, diplomatic solution.

Yet the country's apparently prevailing combative mood about the crisis, despite warnings of caution by some politicians and media commentators, was evident in the warm send-off given to the aircraft carriers Invincible and Hermes as they left Portsmouth's historic naval base in warm spring sunshine with Marine commanders, fighter planes and combat helicopters aboard to lead the Falklands-bound British task force.

The fleet also will include destroyers, submarines, frigates and support ships, many of which are already on their way in the Atlantic. The world's third largest passenger liner, the P&O cruise line's Canberra, also was requisitioned by the British government today to carry troops and hospital facilities outside any combat zone. Government officials said more ships like the Canberra, which was diverted from carrying 1,500 vacationers on a Mediterranean cruise later this week, may be pressed into military service.

With the British armada two or three weeks away from the Falklands, where the Argentine military reportedly is continuing to reinforce its position, British and American diplomats here said they were searching for ways to increase pressure on the Argentine government to withdraw and negotiate.

International economic sanctions against Argentina and a sea and air blockade of the Falklands are among the options being considered, according to the diplomats. Although it would take time, they said it was possible for the British task force to eventually cut off the Argentine troops on the arid, windswept Falklands from military supplies, food and even water.

The British government, which already has frozen Argentine assets here, is sounding out the other nine members of the European Community and allies like the United States on tough economic sanctions, despite West Germany's extensive and lucrative trade with Argentina and U.S. treaty ties through the Organization of American States. British citizens were advised today by the government not to travel to Argentina, and the estimated 17,000 Britons living in the South American country were advised by a British Broadcasting Corp. radio World Service broadcast to leave.

Despite President Reagan's failure to talk Argentine President Leopoldo Galtieri out of invading the Falklands, the Reagan administration is still seen here as crucial to diplomatic efforts to defuse the crisis.

Carrington's decision shocked British and American diplomats here, as well as many senior politicians in the governing Conservative Party. A charming but sometimes aloof aristocrat, Carrington was widely respected in diplomatic circles, including by U.S. Secretary of State Alexander M. Haig Jr., despite public disagreements about Middle East policy.