Yes, it should by modern language standards, but no, that's not how the official name of the country is written.
You'll see many older names lacking the definite suffix, and this is a result of the strong Danish influence on the Norwegian language at the time. Since they're proper nouns, they're seldom changed to conform with current language norms.
In most settings, the country will be referred to as "Mexico", plain and simple.
Yes, it's confusing, but yes, there's a reason for it.
The "x" in "Mexico" is preserved because it's a proper noun, and we try not to mess too much with proper nouns - just like you wouldn't change the the spelling of someone's name based on your own pronunciation. "Meksikansk" is just an adjective, and we have less qualms with changing those to conform with Norwegian orthography.
You'll see the same thing with "Canada" and "kanadisk". Curiously, both "kanadisk" and "meksikansk" still have optional spelling variations preserving the "c" and "x" respectively - but these forms are much less used than the ones I've listed.
the constitution of the country refers to the country as 'los Estados Unidos Mexicanos'
It's understandable but wrong assertion. But people, 80 excess down votes? I know we're all right and poor kerrie 73 made a mistake, but come on?
Kerrie73's objection is perfectly understandable. After all, we native born English speakers are certainly not used to seeing Mexico referred to this way. I've lived in North America all of my life (40+ years) and never referred to or heard of Mexico referred to this way in daily life until now.
So I'm giving Kerrie an upvote and a lingot just because I feel bad for him or her running afoul of this "crowd" mentality. I encourage any of you with a sense of the fact that we're all in this together as language learners and sharing our various global perspectives to do the same! Have a great day!