No, astronauts from many different nations have travelled into space, and NASA will accept applicants who were born outside of the United States provided that they have US citizenship.
Alan Shepard was the first US astronaut and the second person in space.
The last US astronaut to go to space alone was John Young. He flew a solo mission aboard the Gemini 10 spacecraft on July 18, 1966.
Philip Kenyon Chapman was the first Australian born American astronaut (NASA Group 6; 1967). This question can be answered in two ways, Philip Kenyon Chapman is the first Australian born (he gained US citizenship in 1966) to train as an astronaut, although he never flew in space. He was the mission scientist for Apollo 14. Paul Scully Power is an Australian born Payload Specialist who flew aboard STS-41 in October 5th to 13th 1984, but by that time he had become an US citizen, which he took in 1982. So while Chapman holds the record for the first Australian to train as an astronaut, Power holds the record for the first Australian in space.
Barbara Morgan, who was a teacher turned astronaut, is from Idaho. She was born and raised in Fresno, but considers McCall, Idaho her hometown.
Sunita Williams is an American astronaut, she was born in Ohio. She is of Indian and Slovenian descent, her father being the former and her mother the latter.
He was born in Ohio on August 5, 1930.
US astronaut James A. Lovell Jr. was born on March 25, 1928.
That depends on what type of astronaut your talking about such as pilot: which would need a bachelors degree in engineering
Brian Duffy - astronaut - was born on 1953-06-20.
Gary Payton - astronaut - was born on 1948-06-20.
Steven Smith - astronaut - was born on 1958-12-30.
Stephen Bowen - astronaut - was born on 1964-02-13.
Steven MacLean - astronaut - was born on 1954-12-14.
Michael Barratt - astronaut - was born on 1959-04-16.
John Young - astronaut - was born on 1930-09-24.
Scott Kelly - astronaut - was born on 1964-02-21.
Buzz Aldrin