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Not unless you're a black man with a white woman

It doesn't explicitly say it but it's been interpreted that way and upheld to the same effect a couple of times. What section 1 (the passage in question) of the 14th amendment actually says is thus:

All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.

Basically, the prohibition on the abridgement of rights by a State (as in, one of the US 50 states, not the classical political definition of a sate) means that one state can't prevent you from entering another state. If you can exist in one state you can exist in all of them, and none of the other 49 can say anything about it, and it follows therefore that if you can exist in any state none of them can do anything to keep you out.

This has also been taken to mean that one state can't prosecute somebody for something done in a different state. I.e. New York can't prosecute someone in Florida for violating New York's vaccination laws. It seems at first blush that New York could indeed prosecute somebody if they actually physically went there, but that's a thorny legal issue that calls into question lots of other already established precedents. For instance, the fact that all 50 states recognize each other's drivers licenses and license plates as valid, even though the actual requirements for obtaining either can vary widely between states. California probably really hates that you can drive a car that's not CA smog compliant into their state from, say, Alabama. But ultimately there's nothing they can do about it since the vehicle was indeed legally registered where its owner lives. You couldn't register it in CA but CA can't stop you from registering it in AL.