Watch this brief video of Marco Rubio on CNN concerning making English the official national language of the U.S., then see my comments below:
Concerning whether English could be made the official language of America and be constitutional, I believe there are only three ways this can be done.
1.) Through Congress's authority to make a"uniform rule of naturalization" in Article I, Section 8 of the Constitution.
2.) Through a constitutional amendment.
3.) Through their authority to regulate immigration, which I argue comes from Article I, Section 8, Power number 9, which states that Congress shall have the power "To define and punish piracies and felonies committed on the high seas, and offenses against the law of nations..." (See here for further explanation on why I argue this).
Since passing a constitutional amendment making English the official language of the U.S. would be the final say on the matter, thus binding all branches of government, including the judiciary, this would be the most direct route, yet also most likely the most difficult route to take, given the difficulty of passing a constitutional amendment, either through Congress or via state conventions. Thus, I will not comment any further on the matter of amendments.
However, with regards to whether Congress can make such a law that establishes English as the official language of country, I believe Congress could find authority to make English the "primary" language of the country (but not the "official national language") in their power of making a "uniform rule of naturalization," making English classes and a final competency level in English required for the naturalization process, perhaps being measured by some kind of test like the Michigan Test or TOEFL. They might even perhaps be able to limit immigration into America to only those who speak English at a competent level, not that I would agree with such legislation.
This would be the closest Congress could get to establishing English as an official language, but I do not think Congress has authority to pass outright a law defining English as the official national language, which authority would remain either with the states themselves or to the people, via the 10th Amendment. Congress can merely make English the primary language of the nation in practice by requiring all residents hoping to be naturalized to citizenship to be competent in English or by limiting immigration to only those immigrants who can demonstrate competency in English. My opinions on such actions will remain with myself, but I think Congress could bring about an unofficial "national language" of English via these 3 means.