President Carter lost his last significant Republican support for election-day voter registration yesterday when House Minority Leader John J. Rhodes changed positions and announced he will vote against the legislation.

The switch means the legislation faces party-line opposition when it comes up in the House next week. Supporters said this means a close vote, since some Southern Democratic defections are expected.

Rhodes (R-Ariz.) originally backed the proposal as "a good idea" but said mandatory identification of election-day registratns, such as showing a driver's license, should be requires mandatory identification but the House measure, in its present form leaves this decision to states.

however, Rhodes, in a statement yesterday, raised points never mentioned in his original statement of support. He called for making universal voter registration optional for the states and said failure to do so was "a dangerous step toward the federalization of the entire election process."

Rhodes has been under unremitting attacks from party conservatives for his original stand. A recent cartoon in the newsletter of the Citizens for the Republic, the Ronald Reagan political action committee, shows a bleeding GOP elephant that has been stabbed by a figure labeled "Rhodes" and "universal registration." The caption says: "Et tu, Rhode."

The reaction of Rhodes' critics is based on the belief, widespread in both parties, that election-day registration would benefit Democrats. Rhodes took the view that Republicans should put themselves on the side of "opening up the process" rather than being seen as opponetns of greater participation.

Rhodes said yesterday, without offering specifics, that he still believed this and had not "abandoned my search for a reasonable and fair plan to broaden voter participation."

Following Rhodes' lead, the House Republican Policy Committee yesterday adopted a statement of formal opposition to the legislation.