"did I spell your name correctly?"
=> (내가) 네 이름 맞게 쓴 거니? (casual - between friends) * 이름 = name.
=> 제가 (선생님/사모님/님) 성함을 맞게 썼나요? (formal and respectful) * 성함 = name (honorific).
Above sentences are using 맞게 (correctly), which seems to fit the situation best.
If you want to emphasize "spelling", you could say:
- 네 이름 철자[스펠링]를 맞게 쓴 거니? / 성함 철자[스펠링]를 맞게 쓴 건가요?
철자 means "spelling", same as 스펠링 which is borrowed from English.
Nowadays 스펠링 seems more widely used than 철자.