The vote was 109-28, with two delegates abstaining.
A more litigious approach to the conflict was endorsed by the Senate last month, raising the possibility of a showdown on a racially charged issue between the two legislative chambers before the session ends Monday night.
The House bill would let independent mediators decide whether the state's 2005 decision to grant Towson a master's degree in business administration violated Maryland's desegregation commitments. Morgan State University officials contend that decision violated federal and state civil rights laws because the degree was already available at nearby Morgan, a historically black university.
The Supreme Court has generally discouraged such program duplication on the grounds that it makes it easier for white students to avoid enrolling at black campuses.
The Senate version of the bill, which passed last month, would allow Morgan to take the matter to a state court and ask a judge to dismantle Towson's MBA program. House leaders said they preferred a path of mediation and binding arbitration over the unprecedented prospect of public colleges suing their own state over academic programming.
The Senate bill's lead sponsor, Baltimore Democrat Joan Carter Conway, has said she would likely fight the House's amendments. A legal matter ought to be settled in a court of law, she said.
Gov. Martin OMalley has said he is inclined to support the Senate's approach to the conflict, but both chambers would have to reconcile their different versions.