The United States is a democratic republic under the rule of law. Or, the United States is a representative democracy under the rule of law.
The officials chosen by the people to govern are bound by the laws, and the powers of government in general are constrained by the constitution. Insofar as the government acts within the constraints of the constitution, it does so in accordance with the will of the people, as expressed through elections. For example, the constitution gives the government the power to regulate foreign trade by enacting tariffs. If a majority of the people decide that the tariffs are too high or too low to suit their interests, they may elect legislators who pledge to lower or raise them, as the case may be. Because there are certain subjects the U.S. Constitution declares to be beyond the compentance of government–for example, making laws infringing on the freedom of speech–then even a majority may not simply elect legislators who will pass laws infringing on the freedom of speech of unpopular minorities. Note, however, that under the U.S. Constitution a sufficiently large majority can in theory change just about anything–if a large enough majority of the people want to, and want to intensely enough and long enough to get a constitutional amendment through the process of being approved, they could alter or do away with, say, the Second Amendment, or the Establishment Clause of the First Amendment, or the Thirteenth Amendment. In practice, there is for the most part a strong extra-constitutional cultural commitment to not abridging the Bill of Rights. Of course, no constitution will function if enough people simply decide it’s no longer valid; consider the fate of the Articles of Confederation.
I don’t think you can just reduce “republic” to the “rule of law”, by the way. It’s at least possible to conceive of a monarchy where the king is selected strictly by hereditary succession and rules for life, but in theory must rule only in accordance with the laws and traditions of the country. In fact, historically many if not most monarchs have been in practice constrained by the traditional laws and customs of the countries they have ruled. You can obviously have a non-democratic republic, in which a minority of the population elect those who govern in accordance with the constitution and laws (e.g., South Carolina in 1860). Theoretically you could have a pretty lawless republic, too: all free citizens (however defined) elect a legislative assembly and a separate executive with veto power, all for fixed terms, but so long as the executive and legislative branches agree they have totally sovereign power to do whatever they want–proclaim Unitarian Universalism to be the state religion, abolish private property, end all marriages and institute universal “free love”. Then, when the citizens tire of this, at the next election they put in different legislators and a different president who institute strict Islamic law. And round and round they go.
In practice, when Americans talk about spreading “democracy”, they usually mean this as a shorthand for a system in which the legitimate powers of government are structured by some sort of constitution to prevent abuses (or just general silliness like the above), and also in which the rights of individuals are guaranteed against the power of even the majority in certain specified areas; for example, even the majority has no power to compel the adherents of even the tiniest and most unpopular religious sect to renounce their gods. However, such a system of rule of law is coupled with the understanding that the will of the majority will prevail in those areas where the government may legitimately act: if the people want highways to be built, then highways will be built, even though the minority wanted railroads, or canals, or bike paths. We don’t want a system where the wise and benevolent philosopher-kings rule us “for our own good”, both because such systems (human nature being what it is) tend to wind up being ruled for the good of the philosopher-kings (or aristocrats, or Party members, or members othe Herrenvolk) rather than that of the people as a whole; and also because the citizens of a democratic republic aren’t children, and should be permitted–within the limits of the constitution’s guarantees of fundamental individual rights–to govern themselves as they see fit.