Would legalizing all drugs in the U.S. end the Mexican Drug War coming into our country?

No, it would increase it by a factor of ten.

Mexican cartels aren't fighting each other for political or ideological reasons, they're at war over money.  The black market is just like any other business market:  People want good products at a reasonable price, and they'll pay cold hard cash to whoever can provide both.  

The cartels and their supporters are killing each other (and anyone else who happens to wander within range) in record numbers, and they're doing it in order to control the flow of a product that isn't even legal to buy in America.  If those products were suddenly legalized, now we're talking about really big money on the line.  The money from a legalized drug market would make the money they're making now look like pocket change.

Minus the US market much of it might dry up. What that would do to the social fabric of this addicted nation is yet to be seen. But we have a test case in Oregon right now so hopefully that'll be studied to see if the US has the self control to manage legalizing narcotics as several other nations have done. Personally I somehow doubt we can be trusted with it.