Before you get to international law issues, it is uncommon for a state to have a law against murder that applies outside of the state's territory.
For example, in 2003 it was alleged that a US citizen (Tina Watson) was murdered by another US citizen in Australia. In order to try the alleged offender in the United States (specifically in Alabama), the prosecutor had to prove that there was a conspiracy and that part of the conspiracy was carried out in the US. (They were not able to so prove.)
Since 2002, Australia has had a law that says that if you murder an Australian citizen anywhere in the world then you can be tried in Australia: Criminal Code (Cth), s 115.1. This law was enacted in response to a terrorist attack in Indonesia in which many Australians well killed. This law might apply if the US executed an Australian citizen. There is a defence in the Commonwealth Criminal Code for lawful authority (s 10.5) but the word "law" is defined in the Code's Dictionary to refer to Commonwealth law, not foreign law.
The Australian authorities would then look at the international law and political issues discussed in the other answers; political discretion in a s 115.1 prosecution is maintained by the requirement for that the Attorney-General's consent is required for such a prosecution: s 115.6.
France's penal code applies worldwide to any offence where a French national is a victim: Penal Code, art 113-7. The French Code has a general defence of lawful authority: art 122-4, but I don't know French (or French law) well enough to say how a French court could would look at a US law permitting capital punishment.