Not necessarily. A republic is simply a country without a monarch. China is a republic. Iran is a republic. Russia is a republic. Democracies that are not republics include places like Denmark, Netherland, Sweden and Spain.

The canard you're bringing up stems from a weird misunderstanding of what republics and democracies really are. Democracies have laws, the same rules for everyone, they have constitutions that (try to) guarantee equal rights. Democracies often have better "rule of law" than many republics, but ultimately, they believe that the authority stems from the people and that the government serves the people rather than the other way around.

Fortunately many republics are also democracies. France, Germany, and, as many people would really, really like to believe, the United States of America. The fact that so many Americans don't really care about democracy, undermine its legitimacy with misleading arguments, and support an electoral system that denies minority representation while letting the loser of the popular vote win the presidency, is really sad.