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.
William Taylor - police officer - was born in 1947.
Thomas Arnold - police officer - was born in 1835.
John May - police officer - was born in 1775.
Chester Williams - police officer - was born in 1973.
John Gerrard - police officer - was born in 1920.
Frank Elliott - police officer - was born in 1874.
Bill Hughes - police officer - was born in 1950.
Douglas Webb - police officer - was born in 1909.
Rafael Pérez - police officer - was born in 1967.
Patrick Kavanagh - police officer - was born in 1923.
John Grieve - police officer - was born in 1946.
John Maxwell - police officer - was born in 1882.