President Carter's extraordinary appearance before the Democratic National Committee Friday was a confession that the domestic issue that has preoccupied him since he took office - energy - has more than ever brought his own leadership abilities into question.

The day after House members of his own party voted overwhelmingly to repudiate his policy on the decontrol of domestic oil prices, the president delivered a lecture to the DNC in the stern tone of a gravely disappointed but still loving parent.

The speech and a followup question-and-answer session was one of Carter's most revealing appearances for it reflected all of the emotions that have been welling up inside the White House during the last few weeks as the Democratic-controlled Congress has remained deadlocked and divided on the question of energy and its political twin, inflation.

Those emotions include frustration and anger with congressional inaction, particularly on the question of oil, and despair tinged with self-pity that the Congress, buffeted as Carter said Friday by "hundreds of well-financed and powerful special interests," will ever agree on any energy or inflation policy, let alone those proposed by the administration. Finally, those emotions included fear about what continued division and inaction on these issues will mean for the Democratic Party and the Carter presidency in 1980.

What Carter did not say in his appearance is how he, as president of the country and leader of its majority political party proposes to bridge the divisions in the country and the Congress and end, in his own words, the "paralysis, stagnation and drift" he sees afflicting Capitol Hill.

President aides increasingly throw up their hands in despair when asked what can be done about the growing gulf between a Democratic president and a Democratic Congress on the energy issue.

The White House response to that gult has been, in effect, more of the same - a stepped-up schedule of the periodic tongue-lasings of Congress that Carter and his aides have engaged in on the energy issue for almost two years.

Convinced that they are right but apparently devoid of ideas on what to do about it, the White House aides have dug in for a protracted struggle with Congress over the energy issue, whatever the political costs.

Those costs could be extremely high. Carter warned the Democrats that who wins and who loses in 1980 will be decided by "how we handle the energy question." The American people, he said, "are interested in seeing [whether] we can work together."

But the next presidential election will be mostly a referendum on Carter's leadership. And the fact that doubts and divisions over energy are posing a growing threat to Carter's leadership was never more evident than at the conclusion of his somber talk to the DNC, when he invited questions. The first questioner walked to a microphone and said:

"You are our public and party leaders and yet, Mr. President, it often appears that you may be reticent to fully exercise the entire prestige and power of those positions to bring about all these solutions which you espouse."

Carter's response to what amounted to an accusation that he has been a weak and ineffectual leader - the likely main theme of his 1980 opponents - was revealing. It included a rare, public admission of failure, a lament about the troubles that have befallen him and, finally, a stubbord declaration that, whether or not his fellow Democrats like his brand of leadership, he will not change it or easily surrender it to someone else o either party.

Recalling the long and frustrating history of his attempts to wine passage of a national policy on oil, the president said, "In April of 1977, I presented to the Congress a comprehensive, reasonable energy proposal. And I have been scorned and ridiculed by the press because I said this was a moral equivalent of war and we actually have a very serious question. In many ways, I have been a lonely voice up until this moment."

But, Carter said, he was not blaming the American people or even Congress for this, because the people still believe there will be a "miracle," that oil "is going to be released from secret hiding places."

He recalled other frustrations, such as his "real wage insurance" proposal for fighting inflation and his hospital cost containment legislation, now buried in congressional committees.

"I admit this failure that I just described to you is, to a major degree, my fault," the president said. "Maybe if I was a better politician, I would have gotten these bills through the Congress."

So what does the president, the party leader, propose to do about these matters - about what he sees as the American people's blind faith in secret oil supplies, about the stalled domestic legislation in Congress and the upcoming Senate debate over the strategic arms limitation treaty with the Soviet Union?

"I have done the best I could," he said. "I have never backed down. I am going to continue to fight . . . And I haven't made my announcement of what I am going to do in 1980, but I have never backed down from a fight, I have never been afraid of public opinion polls and if and when I decide to run, it will be in every precinct in this country, no matter who else runs, and I have no doubt that it will be successful."

From the beginning of his presidency, Carter has repeatedly declared his intention to fight for what he believes is right, whatever the results. Another declaration of that intention is not likely to be any more effective than the earlier ones, but it is what the president offered his fellow Democrats.

The message seemed clear: Like it or not, the Carter White House will remain on its present course, even if it means further estrangement from the congressional wing of the president's own party.

The message to disgruntled Democrats who long to nominate Sen. Edward M. Kennedy (D-Mass.) - who leads Carter in virtually all of the public opinion polls the president said do not frighten him - seemed equally clear. When an incumbent president, not yet an announced candidate for reelection, declares that he will run "in every precinct in this country," he is warning those who would take the nomination from him that the prize may not be worth the cost of the bloody battle.

Like it or not, Carter was telling the Democrats, they will sink or swim with him in 1980.