Georgetown has several fraternities and sororities that are not recognized by the school. However, there is definitely not a state school/southern type of greek atmosphere. However, if you choose, there is the opportunity for you to join a fraternity or sorority.
I am a sophomore and member of Delta Phi Epsilon Professional Foreign Service Fraternity. It is the largest fraternity on campus and it also has a sister sorority which is somewhat less popular. http://www.deltaphiepsilon.net
Below I have copied the greek life section of the wikipedia article on Georgetown that I helped edit.
Georgetown University does not recognize the existence of fraternities, sororities, and secret societies among the student body. Georgetown's Student Affairs Policy specifically prohibits "2. Fraternities and sororities: single sex groups with ritualized, demeaning or secret membership practices, and specifically those organizations affiliated with the national Intrafraternity Council, Pan Hellenic Association, and Pan Hellenic Council. 3. Secret societies: groups that do not disclose their purpose, membership or activities, or whose purpose, membership or activities are discriminatory." from receiving access to university benefits.  Many students are not aware of their existence either, as fraternities and sororities enjoy only limited visibility.
Fraternities with chapters active on campus are Delta Phi Epsilon (professional) (DPE) (Georgetown's Chapter of this professional professional foreign service fraternity, Alpha Chapter, was established in 1920. Its members include several deans of the Walsh School of Foreign Service, as well as Jesuits), Alpha Epsilon Pi (Georgetown's chapter, Eta Sigma chapter, affiliated with campus Hillel, was established in 2002, making it the school's first social fraternity), and the Georgetown University Sigma Phi Epsilon SEC chapter. (Established in early 2005, the chapter is scheduled to receive its charter in Fall 2006). There are also a number of minority interest fraternities chartered at other unversities that have membership on campus. African American interest: Kappa Alpha Psi (American University), Alpha Phi Alpha (American University), Phi Beta Sigma (Howard American University). Latino interest: Lambda Upsilon Lambda (chapter jointly founded at Georgetown, George Washington, and American Universities).
Co-ed fraternities at Georgetown include the business fraternity Delta Sigma Pi (charter revoked 2006), Phi Alpha Delta Professional Pre-Law Fraternity, Alpha Kappa Psi Professional Business Fraternity, and Alpha Phi Omega (APO). Of these fraternities, Alpha Phi Omega is the only fraternity recognized and given funding by the university, as it is seen as a service organization. 
Sororities also play a role in campus life. Some sororities present are Delta Phi Epsilon Foreign Service Sorority, and Lambda Pi Chi (American University) Latina interest, Delta Sigma Theta (Howard University) African American interest and Alpha Kappa Alpha (Howard University) African American interest.