No the employer can not bar dating. The D.C. Circuit has reversed the NLRB's finding that a broad no-fraternization policy was not an unfair labor practice. In Guardsmark, LLC v. NLRB(February 2, 2007), the D.C. Circuit reviewed the Board's findings on three separate rules. The court enforced the Board's findings that the first two rules--which prohibited guards from complaining to client representatives and prohibited solicitations and distributions while off-duty--violated the NLRA. The court also held, contrary to the Board, that the third, no-fraternization rule was unlawful. The court found the whole policy unlawful the board found 2 of three points unlawful.
An employee that suffers an adverse employment situation,may have cause of action in federal court,and can definitely proceed to the NLRB under a Sec.8 unfair labor practice. The case is too new to see if the federal courts will view this broadly.
Employees' ability to interact with each other is often a vital part of their right to engage in collective activity, so this was a welcomed development.