The book was (eventually, at least) huge best-seller...but does that mean someone with no real interest in babies or childcare would have paid much attention to the author?
Roddenberry had two children at the time, Darleen and Dawn, born in 1948 and 1953 respectively. Dr. Spock's famous book was published in 1946 and rapidly became a best-seller. However, keep in mind that this was an era when child care was seen largely as a woman's province, and that Roddenberry was working as an airline pilot until 1949 and a police officer thereafter and might not have spent that much time involved with the raising of his daughters. So while it's likely that his wife Eileen was familiar with Dr. Spock, Gene himself might have only been peripherally aware of the man. By 1964, when he was looking for a character name and thought of "Spock," he might not have consciously remembered coming across the name before.