Here's a 1920 census that lists Agnew's father as an alien.
Spiro was born in 1918.
Fourteenth Census of the United States
Line 72.
1. In 1920, the census taker wrote down that Theodore was an alien who came to the US in 1887.
2. In 1930, the census taker wrote down that he was a naturalized citizen who came to the US (if thats not the date of naturalization) in 1903.
In 1910, Theodore S. Anagnost (Anagnostopoulos) lived in Schenectady, NY, with and next to relatives (cousins, siblings). See dist. 7, enumeration dist. 186, ward 7, p. 4A.
That document states that he came to the US in 1902 and was naturalized by 1910. So perhaps he was naturalized in 1903, shortly after arriving in the US.
Also, if he already was in the country in 1910 and had served in the military and married an American citizen, it would be very odd if he had not achieved citizenship by 1920.