When an adult American citizen dies while vacationing in Mexico with no will but assets in both Mexico and the US, which countries' probate law applies?
The default choice of law rule is that intestate succession is governed by the law of the place of domicile of the decedent at death (i.e. by the law of the state where someone resides in the U.S. in this case), if there is no will and if no other consideration applies.
Incidentally, the citizenship of the decedent is pretty much irrelevant. You don't need to be a a citizen of a place to be domiciled there. Similarly, where you happen to be when you die is also irrelevant to succession although other post-mortem processes like inquests are affected.
Real property, however, is generally governed by the laws of intestate succession in the place where it is located. This requires a separate Mexican succession process, which would be called an ancillary probate in U.S. practice, but probably has a different name in Mexico because Mexico has a civil law legal system that follows the Spanish tradition for succession at death rather than the common law procedural process concept of a probate proceeding, and is often handled by a legally trained Mexican notary outside the court system. At the same time, common law legal systems give near absolute discretion to the testator in how they make their bequests subject to minimum immediate family support rules, while civil law legal systems are usually more limiting.
Intangible personal property (e.g. a bank account) is usually deemed to be governed by the law of the place of domicile.
Whether tangible personal property is governed by the law of the place of domicile, or by the law of the place where it is located at death, isn't an issue that is resolved uniformly in all jurisdictions. It would depend upon how the issue presented itself and in what legal forum it presented itself. Often, the reality that "possession is 9/10th of the law" and that tangible personal property often has little significant economic value, means that these issues are resolved without resort to the courts or formal legal process. (In England, in the early modern era, succession to tangible personal property was vested in the clergy and courts only handled succession to real property and intangible property.)
Also, in terms of choice of law, in the U.S., probate is a matter of state law and is subject to a case law exception to federal jurisdiction that prevents it from being litigated in federal courts. Mexico also has a federal system, but I don't know whether Mexico's laws on intestate succession are state laws or federal laws (I believe that it is governed by state law but that there isn't much interstate variation).