No, despite what this AL.com headlines (and some other sources say as well). The amendment would largely just restate that courts may not use foreign law in those cases when “doing so would violate any state law or a right guaranteed by the Constitution of this state or of the United States.”
Normal American “choice of law” principles often call for the application of foreign law in cases that involve foreign transactions, for instance in some tort cases arising from injuries in foreign countries, determining the family status of people who were married or adopted children in foreign countries, and more. American courts wouldn’t enforce foreign rules that violate Americans’ free speech rights, equal protection rights, and so on; but in the great bulk of cases in which foreign law would be applied, there would be no such constitutional problem. The Alabama amendment wouldn’t bar use of foreign law in such cases.
Nor would the amendment bar Alabama courts from consulting the way foreign countries do things in developing Alabama common law or constitutional law (though I don’t know how often Alabama courts do that). And it wouldn’t bar courts from using religious law, such as Sharia, in those rare situations where American courts may constitutionally use such law, chiefly when they have to apply the foreign law of a country that incorporates Sharia into its legal system.
The one area in which the amendment might bar use of foreign law has to do with contracts that provide that they are to be interpreted under foreign law (a common provision in contracts that deal with foreign transactions). Provision (g) of the amendment doesn’t just bar violations of state law or of the state or federal constitutions but also says, “no Alabama court shall be required by any contract or other obligation entered into by a person or entity to apply or enforce any foreign law.” That might mean that choice-of-law provisions in contracts would be invalid in Alabama courts, because they purport to require Alabama courts to apply or enforce foreign law. But it might alternatively mean that such contracts can’t require Alabama courts to apply foreign law, though Alabama courts could choose to do so following Alabama choice-of-law rules. This will take some litigation to resolve.
And the amendment might also have an effect quite unrelated to foreign law: Provision (g) also says,
Nothing in this amendment shall be interpreted to limit the right of a natural person or entity of this state to voluntarily restrict or limit his, her, or its own constitutional rights by contract or specific waiver consistent with constitutional principles. However, the language of any such contract or other waiver shall be strictly construed in favor of preserving the constitutional rights of the natural person in this state.
This seems to suggest that even a purely domestic waiver of constitutional rights by an individual — most often, a waiver of the right to speak (i.e., an agreement not to say certain things) or a waiver of the right to trial by jury — “shall be strictly construed in favor of preserving the constitutional rights.” Alabama law already appears to strictly construe waivers of civil jury trial rights, but I’m not sure that’s so as to waivers of free speech rights, which are common in various sorts of confidentiality agreements.