He lives in New York City with his long-time partner, painter William Kennon.

[2] When Greenspan was awarded the Herb Alpert Award by the California Institute of the Arts in 2002, the citation called him "A classicist in experimental clothing" whose "plays ask big questions about history, creation, sexual behavior, the complications of family and the very act of performing a play.

"[3] In 2009 he collaborated with Stephin Merritt of The Magnetic Fields in a musical adaptation of Neil Gaiman's Coraline, under the direction of Leigh Silverman.

In an interview with Lizzie Olesker in The Brooklyn Rail, Greenspan describes the musical: "We suggest things.

"[4] In 2022, Greenspan was included in the book 50 Key Figures in Queer US Theatre, profiled in a chapter written by performance scholar Nick Salvato.