That is not true. There is no requirement that an immigrant have any English skills and many do not.
Huh?
"To be eligible for naturalization, you must be able to read, write, and speak basic English."
http://uscis.gov/graphics/services/natz/natzsamp.htm
Alicia Alvarado presses two for Spanish. Nearly everything the 44-year-old janitor needs can be had in her mother tongue. She shops in Spanish, files police reports, talks to nurses and teachers and voice-mail systems. For 17 years, Alvarado has lived in a Spanish-speaking world, first in Los Angeles and now in Concord's Monument Corridor. Her continued monolingualism stems from cultural pride, a hectic life and the challenge of studying a foreign language.
Nathaniel Hoffman in the Contra Costa Times -- 3/25/06