You say that
Celia feels incomplete to you as a first name, but you'd use it as a nickname for
Cecilia, which indicates that you must like
Celia to some degree.
I once had the same problem with
Lucy. I loved
Lucy, but didn't like
Lucille.
Lucy felt incomplete to me as a full name. I kept thinking that it had to be a nickname for
Lucille. So I kept thinking that I'd never use it, because I don't like
Lucille and I wouldn't use a name that I don't like just to get to the nickname.
Then it was pointed out to me that
Lucy is indeed a legitimate name in its own right. It didn't start out as a nickname for
Lucille.
Lucy is English for
Lucia, the feminine form of
Lucius, meaning "light".
Lucille is from
Lucilla, a diminutive of
Lucia.
Lucy actually predates
Lucille, and
Lucille, although it's longer and sounds more formal, actually comes from a diminutive, which
Lucy does not.
In the case of
Celia and
Cecilia, the two names are actually unrelated, so
Celia certainly stands as a name in its own right.
Celia is the feminine form of the
Roman family name
Caelius, from Caelum, and means "heaven".
Cecilia is from the
Roman family name
Caecilius, from Caecus, meaning "blind". There's no need to see
Celia as incomplete.
Of course, you say that
Celia "feels" incomplete, and facts don't always change feelings, so this might not change anything for you. But I know that in my case, facts did change how I felt about
Lucy, so I just wanted to throw this out there. Also, I want to encourage the use of
Celia as opposed to
Cecile or
Cecilia, just because I like it more.