Yes.

He's had several girlfriends, including Julie Madison and Linda Page (very early), Vicki Vale and Silver St. Cloud ('60s/'70s). The Earth-2 Bruce Wayne actually married Selina Kyle. Of course, these were all pre-Crisis (major change in the DC universe) so they probably don't count in current continuity.

Bruce has had several girlfriends throughout the years. His first girlfriend was Julie Madison, "Detective Comics" #32, October 1939, 5 months after Batman's debut in "Detective Comics" #27 and she predates even Catwoman. Now although she has never been classified as a girlfriend for the modern Batman, Selina Kyle was the next "romantic" interest for Bruce and was introduced as Catwoman in "Batman" #1 in Spring 1940 along with the Joker, about a year after Batman's debut and about one month after the debut of Dick Grayson aka Robin in "Detective Comics" #38 April, 1940. However the Earth-2 Batman did end up marrying the Earth-2 Catwoman and the union resulted in the first female hero to use the name of the Huntress. Perhaps Bruce Wayne's most famous girlfriend and the only one to appear in the Batman serials of the 1940s as well as the first feature film was Vicki Vale who first appeared in "Batman" #49 Oct/Nov 1948. Another "romantic" interest, though hardly a girlfriend was Pamela Isley aka Poison Ivy. Though not a girlfriend there has been sexual tension between the two since her introduction in "Batman" #181 June 1966, which was five months after the debut of the "Batman" television series that first aired on January 12, 1966. Talia al Ghul was another big "romantic" interest for Bruce, and one of the few who knows Bruce and Batman are one and the same. In fact Talia, who debuted in "Detective Comics" #411 May 1971, may even have (as of July 26, 2006) given birth to a son of Bruce named Damian Wayne, and since Bruce did not dispute the possibility we know that it is at least a possibility that Bruce is the father of the boy. Bruce dated Silver St. Cloud shortly after her debut in "Detective Comics" #470 June 1977. Julia Pennyworth, Alfred's daughter, briefly dated Bruce when she debuted in "Detective Comics" #501 January 1980. In "Detective Comics" #529 August 1983 Natalia Knight aka Nocturna was introduced and she briefly dated both Bruce and Batman without linking the two together. In the pre-crisis universe she even adopted Jason Todd aka Robin, and she has also been seen in the post crisis universe. Introduced in "Detective Comics" #576 July 1987 was Rachel Caspian, who also dated Bruce, but became the 3rd Reaper in "Batman: Full Circle" 1991. After Bane was introduced, Bruce met the telekinetic physiotherapist, Shondra Kinsolving, who he briefly dated and who used her telekinesis to heal Bruce's broken back. Another major squeeze of Bruce was Vesper Fairchild introduced in "Batman" #540 March 1997. And the final major babe dated by Bruce was his former bodyguard, who also figured out that Bruce was Batman, Sasha Bordeaux, who was introduced in "Detective Comics" #751 December 2000.