The House voted unanimously yesterday to approve an internal investigation of allegations that the South Korean government tried to buy members of Congress with cash and gifts.

By a vote of 388 to 0, with Rep. Henry B. Gonzalez (D-Tex.) voting "present," the House adopted a resolution requested by its ethics committee to give it additional powers to look into charges that former or current members accepted bribes from South Koreans in return for their votes on aid to the Seoul regime of President Park Chung Hee.

Gonzalez, chairman of the select committee investigating the assassinations of President John F. Kennedy and Martin Luther King, said later he couldn't vote for the resolution because he was bothered that amendments weren't allowed.

"I couldn't vote no, because I think something needs to be done about the charges," he said. "But I stoutly resist the procedure of not allowing full consideration and amendment by the membership."

During a low-key, one-hour debate before the vote, Rep. John J. Flynt (D-Ga.), the ethics committee chairman, assured colleagues that the investigation would be thorough and said it might well cost the entire $530,000 his committee has requested for investigations this year.

Rep. Floyd D. Spence (R-S.C.), the ranking minority member, added that the resolution would "reassure the American public that this body intends to deal with these allegations." He said the "politcally balanced" committee - composed of six members from each party - would "show no favoritisim and tolerate no cover-up," no matter who was involved.

The ethics committe inquiry will parallel a current Justice Department investigation known to be focusing on the dealings between a handful of former and current members and Tongsun Park, a Korean businessman and Washington socialite now living in London.

Gov. Edwin Edwards of Louisiana has acknowledged that his wife accepted $10,000 in cash from Tongsun Park while Edwards was a member of Congress. Former Rep. Richard Hanna (D-Calif.) was a business-partner of Park while Hanna served in Congress. Rep. John McFall, (D-Calif.) said he accepted $4,000 in $1000 bills from Park for his secret office account. Each has denied any wrongdoing.

During the debate yesterday, some members said they felt the House was "picking on" South Korea. Rep. William M. Ketchum (R-Calif.) said he thought the ethics committee also should investigate the propriety of member's trips to Cuba and the Peoples Republic of China, because such trips are paid for in part by those governments.

Rep. Delbert Latta (R-Ohio), said, "I don't think we ought to single out one country." He said he sensed "an insidious movemenjt to cut loose this little country that's been a friend of the United States," but said later that he didn't think anyone in particular was directing such a movement.

Observing the debate from the diplomatic gallery was Kim Suk Kyu, a counselor in the South Korean Embassy's political section.

In a related matter, former Rep. James V. Stanton (D-Ohio) confirmed that he - as a legal representative for the embassy - advised Flynt and Philip A. Lacovara, the ethics committee's special counsel, a few days ago of the South Korean government's willingness to cooperate in the investigation.