Fear of asking girl out - is called ...?

If you want to ask a girl out and have no fear of rejection, you have self-confidence.

If you are scared to ask a girl out, it is probably because you are shy.

It is called love-shyness colloquially. The person is called a love-shy.

It can be applied to women also but it is usually associated with men. The term can cover the fear of any romantic interaction with the opposite sex.

• love-shy men are unable to get girlfriends/wives either because they don't know how, or they are too affraid.

• it means finding it difficult to be assertive in informal situations involving potential romantic or sexual partners. For example, a heterosexual love-shy male will have trouble initiating conversations with women because of strong feelings of anxiety.

[Urbandictionary]

Note: If you want a phobia word, there is pussyphobia in slang but I don't recommend using it because it is too slangy and has other connotations as well.

Ildan has pussyphobia because he is scared to ask girls out or talk to them. [Urbandictionay]

Although, the term is used in this article about a man who cannot approach a girl without becoming a nerve wreck from nyc.barstoolsports but it is not a serious news site. They got it from a Telegraph article and reworded it.

An extreme case can be gynophobia (or gynephobia):

...is an abnormal fear of women, a type of specific social phobia. In the past, the Latin term horror feminae was used. [Wikipedia]

This kind of fear or phobia is also called:

Dating anxieties:

• Dating, by its very nature, is a situation in which two people have not already committed to a permanent relationship. So, for many people, if not most people, dating relationships are experienced as insecure attachments and therefore anxiety producing.

(from psychologytoday.com)

Phobia of dating:

• When you feel shaky or phobic about various aspects of dating or rejection by your prospective partner. Often many issues related to phobia of dating can be resolved when you simply ask your partner what to do.

(form www.phobia-fear-release.com)