I am fascinated by the thought of presidential primaries. They feel skewed towards certain places and certain ethnicities. Let's say that in March of the 1st year of a President's term, Congress passes a law saying that all US states must abolish the caucus system and hold a primary on the first Tuesday of March by the end of January in the next presidential election year. If Congress wanted to do this and it was popular enough in Congress and the President signed it (this only matters if it isn't exceedingly popular), would it go through legally?
First, an unsubtle but often misunderstood point. Congress has the sole power to create Federal law. If Congress passes legislation, that legislation is law by definition, and there are few constraints on what laws Congress can pass.The constitution only lists ex-post-facto laws (laws which outlaw acts after they have already been committed), bills of attainder (laws which create extra-judicial forms of punishment), or any general suspension of habeas corpus. Laws can be vetoed by the President, which Congress can overturn with a supermajority; laws can be challenged in Court and removed if they are deemed to be contrary to constitutional principles. But aside from that, and the short list of proscriptions written into the constitution, Congress can write what laws they like.
There are practical limitations on what legislation Congress will attempt to write. There is no sense trying to craft a bill that will never pass, or survive veto or Judicial review, or that will so enrage voters that it endangers members' reelections. But practical limitations can sometimes be overcome.
So yes, Congress could pass such a law. The constitution itself does not say anything at all about parties or primaries, and Article 2, Section 1, Paragraph 4 states explicitly that:
The Congress may determine the Time of chusing the Electors
so there is no obviously unconstitutional aspect to changing the primary or caucus system to set a consistent primary date. If such a law passed and survived presidential veto it would likely survive Judicial revue, and would ipso facto be legal.
We could debate whether the proposed system is good or workable; I can see advantages and disadvantages. However, under the current political context this type of law seems incredibly unlikely. Political parties use the long primary season as an opportunity to create political tensions that stock party war-chests and motivate voter blocks. It may seem grueling to us — and is certainly exhausting for the candidates — but it is finely-tuned to generate massive donations and support. Shortening the season in the way you suggest would create internal problems for campaigns: less time to gather funds, less time to gather information that enables targeting funds effectively, difficult decisions about where candidates should focus their time and effort... Our current parties are machines geared to a certain utility function that relies on localized monitoring of voter behavior over time combined with targeted fund-raising efforts. Changing that dynamic is is a poiltical risk that everyone involved would resist mightily.