A hard NO. Trey Gowdy would talk a good game for four years and deliver nothing. I was once a big fan of his but became thoroughly disappointed by his lack of action.
What power do you perceive Gowdy had in a Demorat House? All he did was highlight, and that is the extent of what he could do.
Even if the House retained its historical power of arrest, the Speaker would not let that stand, nor would the Sergeant of Arms.
So again, what could have Trey done?
What authority did Gowdy possess that would have allowed him to take whatever action you are referring to?