This is actually a very simple question with a very complex answer. Article 4 Section 4 of the Constitution of the United States guarantees every State in the Union a Republican Form of Government, but that, in and of itself, does not adequately answer the question because we actually have two federal government systems that operate independently and simultaneously in congress.
The Senate is the original confederate assembly of the States as the union, a discrete democracy, all the States are assembled as equals with equal suffrage to reach a majority consensus of all the States as the Union to determine the decisions of a free and independent state; levying war, concluding peace, contracting alliances, establishing commerce, and any other acts or things which an independent state may by right do, which the States as the Union manage through treaties. However, the Senate is also the predominant governing institution in the Confederated (compound) Republic which manages how the States interact with each other to form and maintain the Union through laws. Therefore the Senate, the democratic governing institution, has the power of concurrence over all laws and all treaties, making the democratic assembly the most important, and predominant, governing institution in the United States.
So, does our system of government with the democratic assembly as the dominant governing institution satisfy the requirement of a Republican Form of Government?
The strange answer to that question is yes, because a Confederated (compound) Republic is a Democratic and a Republican Form of Government. Because it is has characteristics of both forms of government. To understand this we need to dissect democratic and republican forms of government to get to their basic principles. “A Democracy” is a legislative assembly, “Democracy” is how that legislature assembly is formed and how it functions to reach a majority consensus of all the members of the democracy. Consensus choice is the legislative process which governs how the legislative assembly is operated. Therefore, we must understand what constitutes “Consensus Choice” before we can understand how a democracy operates.
Consensus choice has two parts to decision making, the first is identifying and ranking choices to form a ballot of the most desirable choices, those top choices are deliberated and the members of the democracy determine the choice by vote, 1 vote per member, and a majority of the members is necessary to the choice. This is the same mode of decision making used in a Confederated (compound) Republic, where the ballot of top choices is determined by the popular branch then the choice is made by the equal branch, whereas in the democratic example the ballot of top choices is made by a quorum of the members then the choice is made by a majority of all the members, the requirement for assent to a measure is the same in both, by vote of all the members, 1 vote per member, and a majority of all the members is necessary to the choice. The only difference is how the quorum is formed and the power of the minority, which is the majority of the popular branch, to block legislation that is not in their interest, which is possible in a Republican Form of Government, but impossible in a democracy, in a democracy there is no power to control the scope and cost by the more populous members, giving the less populous, more numerous, members total control over adopting all measures without the consideration of the most populous members.
This is why the Constitution of the United States was established, to establish a checks and balances legislative system which balances the interest of the largest, more populous and wealthiest members, with the smallest, more numerous members, to form an iterative decision making process which is dependent upon the concurrence of both branches of the legislative assembly. But also notice, the original confederate assembly was preserved in Articles 2 & 3 of the Constitution of the United States, with a higher threshold for assent, 2/3 of the States, to make the decisions of a free and independent State and to manage the general affairs of the government through establishing executive and judicial departments under their direction, as was originally established by Article 9 of the Articles of Confederation, with only minor changes to establish continuity and stability of government through 6 year terms and offset elections for Senators, and to appoint a person to preside over both the Executive and Judicial departments, which allows the Senate, “The Committee of the States”, to recess with the full congress only to be called into session when the consideration of the States as the Union is required.
So the answer to the original question; is the United States a democracy or a republic, is that the United States is a democracy and a democratic/republican form of government, a Confederated (compound) Republic. However, they are separated by the mode of assembly and action, as one is for managing domestic policy through laws, and the other is for managing foreign policy through treaties.