Democrats are pushing to block those on the terror watch list from buying guns.
Lawmakers have focused on the visa waiver program in the wake of the Paris attacks.
The majority leader started the fast-track process for a House-passed bill.
Paul's legislation would require the District of Columbia to grant concealed carry permits.
Sen. Jack Reed (D-R.I.) blocked McCain's request with "great reluctance."
The Justice Department is already looking into the Republican's activities.
Cristina Marcos 11/19/15 01:57 PM EST
The legislation will now go to the Senate, where it may face a tougher path to passage.
The presidential candidate's push thwarted a housing and transportation bill.
Keith Laing 11/19/15 01:14 PM EST
The measure extends transportation funding until Dec. 4.
Cruz and Rubio say they will support nothing but a full repeal.