I created a fighter character, and leveled it up to level 3. When I hit level 3, the DM asked me what my martial archetype would be and I said I didn't care, because I didn't know and knowing him, he wouldn't explain it well. He didn't give me one apparently, and I am level 4 now and have figured out what they are and want one. Now he's not letting me pick one for myself and says it's too late to choose one. Can I or can I not choose an MA even though I'm level 4?
You are not level three or four without a Martial Archetype
If your DM has not allowed for you to pick your Martial archetype, you can not be a level three or four, really you’re still a level two. Even if you’ve increased all of your stats, such as increasing your hit points or ability scores, when levelling up, you have only partially levelled up.
Here is the description of Martial Archetype, page 72 of the Player’s Handbook:
At 3rd level, you choose an archetype that you strive to emulate in your combat styles and techniques. Choose Champion, Battle Master, or Eldritch Knight, all detailed at the end of the class description. The archetype you choose grants you features at 3rd level and again at 7th, 10th, 15th, and 18th level.
Note that it does not say “you can choose”; it says “you choose”. There is no option to not choose a Martial Archetype; it is an imperative.
For context, every class has a subclass they choose when they reach level 3 (sometimes its lower for other classes or certain archetypes but that is not relevant here). Bards have Colleges, Monk’s have Monastic Traditions, Clerics have Domains, Druids have Circles etc and Fighters have Martial Archetypes.
From page 15 of the Player’s Handbook:
Consult the information in your character's class description to see what other improvements you gain at each level.
In the first tier (levels 1–4), characters are effectively apprentice adventurers. They are learning the features that define them as members of particular classes, including the major choices that flavor their class features as they advance (such as a wizard’s Arcane Tradition or a fighter’s Martial Archetype).
This provides evidence that you must learn a Martial Archetype as a Fighter, the book is assuming that you will pick one, there is no option to not pick an archetype. Again, note the lack of the words “can choose” and see how it says “are learning” not “might be learning”, the characters are learning about their subclasses.
For some colloquial evidence, I will use the 5th Edition Basic Rules set. This, as the name would suggest, is a book focusing on only the basic things a new player needs to know in order to play the game. It is official material made by the D&D game designers, meant as a way to make the game more accessible to new players. The Classes section on page 22 says this:
Your class gives you a variety of special features, such as a fighter’s mastery of weapons and armor, and a wizard’s spells. At low levels, your class gives you only two or three features, but as you advance in level you gain more and your existing features often improve.
The fact that even the Basic Rules assume you will pick a subclass and offer subclasses shows that gaining a subclass is a fundamental foundation for your class.
For even more colloquial evidence, the fact that other official game material such as Xanathar’s Guide to Everything, the Sword Coast Adventurer’s Guide and even the Guildmasters’ Guide to Ravnica offer new subclasses shows they have at least some level of importance. If subclasses were not important, Xanathar’s would not have included 31 new subclasses, 3 of those being new Martial Archetypes for the Fighter.
Lots of unofficial material and homebrew material, such as that found on the Dungeon Masters Guild, also includes new subclasses. The fact that even players and dungeon masters of the game are creating new subclasses shows that they are important to the players.
In short, no, you can not pick a martial archetype after level three - purely because by not picking an archetype, you are not level three. Your DM must allow you to pick your Martial Archetype, as otherwise you can’t properly play the game. You will not have the abilities you need to survive harder encounters. Yes you might have increased health and stats, but they just prolong your death; it is your abilities, most of which are gained from your subclasses, that make your choice of class important.
It is unfair to you that other players get subclasses but not you; it is also going to make the game unbalanced against you. If you keep playing as you are, you’ll essentially be a level 2 character with a lot of health when everyone else are level 18 characters.