As Jenna indicates, the word 'public' in the name of a place doesn't give anyone an automatic right to enter. The licensee of a public house can permit or refuse entry to whomever he likes (or doesn't like). The only exception would be that he couldn't do so directly because of illegal discrimination (e.g. on the grounds of race or sex). Similarly, the staff of a public library have the right to refuse entry to anyone they'd rather not have on their premises.
A nightclub (of the type where you simply pay on the door, rather than filling in a membership form) isn't a 'private members' club' but it remains private property and the management have the right to refuse admission to whomever they (don't) like [as long as, once again, they avoid those forms of discrimination which are illegal].
Under certain circumstances, a nightclub might be obliged by law to refuse admission to potential customers. (e.g. when they're under-age, when they're already intoxicated or when the club is already full to it's specified safetly limit). However, there's never any circumstance when a nightclub is obliged by law to admit someone.
Chris