No, "republic" simply means that governance is a public matter, not the private concern of a sovereign. It has more to do with the government's theory of legitimacy than how it is actually governed.

The UK is non-republican because it's governed by Her Majesty's Government, but democratic because the actual power to choose the government belongs to the common people.

The US is republican because the government is constituted by We The People. If it became non-democratic, it would likely remain republican because changing a government's theory of legitimacy is harder than changing how the government actually works.