Can you search for a birth record by birth date only?
No.
The reason why you can't is that actual dates within the year have never been included in the indexes, so it's not a case of the search not having been set up, but a case of the information not being readily available to be searched on.
You can't search by year either without at least one other piece of information - which could be surname
or given name
or county or parish/registration district.