The House is expected to approve the bill later in the day.
Lawmakers face an end-of-the-day deadline to pass a funding bill.
Paul suggested Cruz lacks "personal relationships" with his Senate colleagues.
Asked if he would seek election to any leadership position, Gowdy replied, "I'm not."
The six-month extension is headed to Obama's desk.
The bill would give states flexibility in withholding Medicaid funds.
The Ethics panel is probing whether the Republican improperly helped his wife lobby Congress.
The maneuver allows GOP leaders to bring up a spending bill faster to avoid a shutdown.
House Rules Committee Chairman Pete Sessions is joining the race for majority whip.
Rep. McHenry touted his experience as chief deputy whip.