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.
The measure would require an audit of the central bank.