Pence says the bill sets up an unnecessary barrier to transparency.
The Vice President as well as former House Speaker John Boehner are receiving the Laetare Medal from the university during its upcoming commencement.
The legislation comes after ESPN filed a lawsuit claiming Notre Dame Security Police violated public records laws.
At stake is whether the private university's police department should have to comply with state public records requests.
Three sexual assaults have been reported on Notre Dame's campus in the first week of classes.
Whatever the courts decide, the ruling will impact private universities throughout the state that have police forces.
The rejection comes after the Supreme Court ordered the 7th Circuit Court of Appeals to reconsider its first ruling denying Notre Dame's protest.
The University of Notre Dame plans to offer a variety of programs and courses as part of a new partnership with Kylemore Abbey beginning in 2016.
The 7th U.S. Circuit Court of Appeals is being asked to reconsider its decision, which ruled in favor of the Obama administration's contraception requirement.
Hesburgh, who served as the Notre Dame president from 1952 to 1987, died Thursday in South Bend at age 97.