Ultimately, it will take an act of Congress to ensure that only citizens of the United States can vote.
The Constitution does not directly address the matter, as it was silently assumed that only citizens would ever have that right (and not all citizens, at that.) See Article II sections 1 and 2, which deal with state administration of Federal elections.
As of 1924, all states had banned non-citizens from voting in federal elections, and in 1996, Congress added criminal penalties. Otherwise, voting rights are a matter of state law, not Federal law, with these exceptions:
The 15th Amendment gave African American men the right to vote in 1870.
The 19th Amendment, ratified in 1920, gave American women the right to vote.
The 24th Amendment, ratified in 1964, eliminated poll taxes.
The 26th Amendment, ratified in 1971, lowered the voting age for all elections to 18.
Otherwise, the President simply cannot act to restrict voting rights by executive order, no matter how much we may desire the outcome.