Uhh USA is a republic just fyi...
Discussion
And UK is a constitutional monarchy. Still both countries use democracy as a system of determining its government. These things are not opposites.
In a democratic country, the government is run by the people directly or through elected representatives, with the majority rule prevailing. On the other hand, in a republic, the government is based on a constitution that protects the rights of individuals and ensures that elected representatives make laws within the constraints of that constitution.