The conditions for renting an apartment are up to the landlord. Landlords generally want tenants who will pay the rent, do no damage and not disturb other tenants and neighbours.
Presumably the question concerns apartments in the US since you don't have to be American to rent an apartment in Armenia or Paraguay.
You don't have to be a US citizen to live in the US so if apartments were prohibited to non-citizens there would be even more vagrants and homeless bums loitering in the streets and parks than there are now.
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