Would they call it 'gothic'?

Nope!

Nor would they say they lived in the Mediaeval Period. Gothic et al were tagged on by people after the event.

This from the Etymological on line dictionary

Gothic was used by scholars to mean "Germanic, Teutonic" (1647), hence its evolution as a term for the art style that emerged in northern Europe in the Middle Ages, and the early 19c. literary style that used medieval settings to suggest horror and mystery.

pdr is correct.

Their way of thinking about the time in which they lived as an "age" or "era" came from their religious system. To them, they lived in the middle period between the Incarnation and the Second Coming, the medium aevum or Middle Age, from which we get "medieval" and "Middle Ages."
During the Renaissance scholars and future printers rejected the current way of writing as "Gothic" (what we know of as gothic script) in favour of adopting the writing system used by the Carolingians but which they thought of as "Roman" and therefore better than Gothic, barbarian writing. It is the basis for the way our alphabet looks today (there's a reason why "Times New Roman" is called that).