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I know that in each state the voting system is first-past-the-post, a majoritarian method, but I believe that the federal aggregation, for both the president and the Congress, is approximately proportional to the population of the state.
Does this complex system approximate closer to a majoritarian or to a proportional system?
All states are currently using a first-past-the-post system of voting for national elections and essentially majoritarian. Two states, Nebraska ans Maine, for presidential elections use per-district results for choosing electors and award 2 more to the winner of the state-wide popular vote. All other states use a winner-take-all method of assigning electors for presidential elections.
For House of representatives each member is from a district that is of approximately equal population with each state having at least one district. Each district is a first-past-the-post election. This should get a roughly proportional grouping of beliefs elected, however gerrymandering and number representatives not growing with population growth has prevented that in some scenarios.
The senate is now a statewide first-past-the-post election for each seat, as they are intentionally staggered. Selection of senator was changed in 1913 to be a popular election, previously state legislatures selected senators in any method they saw fit.
You don't seem aware that senators are two per state (fixed in the constitution), regardless of the state population. That's often a lighting rod of non-proportional criticism; see e.g. the first article linked in What's (roughly) the smallest percentage of the US population that through its Senators has successfully blocked a piece of legislation?
senators representing about 11 percent of the population can filibuster a bill or those representing about 16 percent of the population can have a majority.
And there are those who argue the Senate [thus] is a bigger problem than the Electoral College from this disproprtionality perspective:
The Senate is a much bigger problem than the Electoral College [...]
America’s nonwhite population tends to be overwhelmingly in large or medium-sized states. To illustrate, the 10 biggest states (by 2018 Census estimates) all have nontrivial percentages of nonwhite voters, while the 10 smallest states mostly consist of rural, overwhelmingly white states.
While the Electoral College is also winner-take-all at the state level, each state’s representation is much more proportional to population. State are given electoral votes equal to their number of House members plus two senators. House seats are allocated to the states proportional to population. Only the extra two votes contribute to disproportionality [in the Electoral College].
Now it's true that actual majorities in the Senate have not reached anywhere near that 16%. By one calculation, the lowest was 47% (in 2017).
Note however that this is a measure of malapportionment among the two parties that by the design of the voting systems end up competing. In actual PR systems in Europe etc., the number of parties actually competing ends up [much] larger than two. It is however much more difficult to estimate these long-run effects as they would require counterfactual assumptions(*), but they are thought to be significant; see Duverger's law .
(*) For example: how many parties would emerge e.g. if the factions in the Democrat or Republican parties would split and create and their own parties instead of competing in the primaries, which are also majoritarian. Neither party is much inclined to these kinds of splits because it would be an electoral disadvantage under the present voting systems.
As an aside, since Wikipedia doesn't have great article on Duverger’s law...
Although Duverger’s central claim is that social heterogeneity will increase the number of parties only when the electoral system is sufficiently permissive, no existing study actually estimates whether social heterogeneity has a statistically significant effect when the electoral system is permissive. As a result, Duverger’s theory has never been directly tested.
The problem like I said is what is the counterfactual scenario (i.e. what are those sources of heterogeneity that would create more parties in a more "permissive" PR system). One relatively recent (but fairly cited) paper on the topic used ethnic groups as a proxy, and showed that in countries where PR is present they get a better representation.
The USA has a complex voting system, and as a result is neither majoritorian or proportional.
The Congress comprises two houses: The House of Representatives and the Senate.
House representatives are chosen by a plurality in districts with roughly equal population. Gerrymandering is at least tolerated, so district boundaries are often influenced by politics to advantage the party in power. Only the 50 states have voting representatives. Washington DC (700k), Puerto Rico (3m) and various territories have no voting representatives, but may have representative able to attend Congress.
The senate has two (mostly plurality-elected) members per state, regardless of population, so representation is very uneven, with tiny states having disproportionate power. And DC, PR, and territories have no representation at all.
The USA is (I believe) unusual in that the senate must approve all legislation, so that the US senate is much more important than say the UK House of Lords.
It is important when considering the USA, to know that many functions normally considered the role of neutral civil servants, are political appointments, for example judges. Presidential elections use a double plurality, as explained by others.
In summary: The USA has some majoritorian aspects of its electoral system, and no proportional aspects, so it is more majoritiorian than proportional.
while the results can approximate proportional, the fact that you elect specific candidates in each district makes it majoritarian. True proportional allocates a number of seats to each party at an aggregate level, then uses some other mechanism for the members of the party to be chosen to take those seats.
