No, although most states require police officers to be U.S. citizens. There are many who emigrated to the U.S. and became naturalized U.S. citizens.
Rafael Pérez - police officer - was born in 1967.
William Taylor - police officer - was born in 1947.
Thomas Arnold - police officer - was born in 1835.
John Gerrard - police officer - was born in 1920.
Frank Elliott - police officer - was born in 1874.
Alexander Jacob - police officer - was born in 1955.
John Alderson - police officer - was born in 1922.
Ian Johnston - police officer - was born in 1945.
John May - police officer - was born in 1775.
Andrew Reed - police officer - was born in 1837.
Hugh Moore - police officer - was born in 1929.
Mike Boyd - police officer - was born in 1952.