James Madison, who should know, defined a republic as a government with representatives elected by the people. It differed from a democracy not by not having majority rule, but by having representatives. ALL republics as Madison defined them imply majority rule in any representative assembly.
I know history and logic are not right wing strengths, but every collective institution in the constitution proceeds by majority rule: House, Senate, and Supreme Court. They are selected by DIFFERENT majorities of citizens to approximate a workable consensus by hoping to eliminate factions that might dominate one majority.
The only exception is each state elects the same number of Senators even if insignificant in numbers of citizens, a compromise done without principle, but adopted as a price of getting the constitution adopted by all.
Try reading the Federalist Papers, at least, before you pontificate on the subject. And then ask yourself why Jefferson, Madison’s ally, was the first to describe the US in English as a “representative democracy,” a term that did not exist until Jefferson coined it.