A few facts to lay out:
There is no "Mexican" language. Over 250 languages are spoken in Mexico. Of these, Spanish is the most widely spoken and is the language of the government. Most of the other 250+ languages are indigenous languages spoken by members of Mexico's indigenous community. Not a single one of these languages is known as "Mexican" as many of them are vastly different from one another and none have any kind of association with the politics, government or culture of Mexico as a whole.
Mexicans speak a fairly strong dialect of Spanish. A dialect is a form of a language that can be understood by most speakers of another dialect of the same language. If you were fluent in the Spanish of any other country in the world, you would have no trouble understanding Mexican Spanish. There is a much greater difference between Spanish of Spain and Mexican Spanish than there is between American English and British English, but there is still no debate that they are both the same language.
The only challenges to this are presented not by linguists or scholars of the Spanish language, but by nationalistic Mexicans with politically charged motives. They wish to claim their dialect of Spanish as their own language, just as some Belgian speakers of Dutch wish to claim their dialect of Dutch as the "Flemish language." The argument is about politics, not linguistics. Anyone with any knowledge of the Spanish language outside of Mexico would laugh at these ridiculous claims, just as we ridicule people who think we speak the "American language."