Socialization is not the problem for homeschoolers most people think it is and YES there has been research done on the matter. This article by the Washington Times summarizes much of the most recent findings, for more there are more citations at the bottom of this answer. The most recent findings are that homeschoolers typically have BETTER social skills than their mortar and brick school peers.
Your assumption that children that are schooled at home get little to know out-of-home socialization may be based on the belief that socialization requires children to socialize other children. Quite to the contrary, everytime a child is taught about manners, experiences interactions with other people in general (on errands with mom for example) and participates in activities within the community (family, church, broader community) they are getting social interaction. This article on the pros and cons of homeschooling I wrote (after a lot of research on the matter) has a section explaining how a greater variety of social interactions (meaning experience with multiple ages etc) is advantageous. Just scroll down to the "Pros". You might also get a chuckle out of Messy Monday's Blimycow.com vlog titled 7 lies about homeschoolers. This kid was homeschooled and has a great sense of humor about the "myths" he lists while he makes some pretty good points.
That said, they do need a VARIETY of interactions including interactions with people of other races, religions, political beliefs etc. to be as well rounded as possible and as adaptive as possible. There is nothing wrong with a child who is shy or doesn't like large group interactions (like happne in most "extracurriculars" people think of first).
Perhaps you have a quieter child that would enjoy small-group or quieter kinds of activities than you have offered already. Even one on one types of interactions still provide good exposures. I can't imagine a kid who would never want to have a play date. Does your library offer a book club for kids? What about Chess or singles tennis lessons? Honor who you have as a child and know that even in public schools there are kids who are "shy", but do keep trying to find something engaging. Perhaps your child would be more comfortable with something the two of you can participate in something like "Indian Princesses" which is a lot like girl scouts but stresses father/daughter activities (I don't know a whole lot else about it, my niece and brother in law are in this club together).
Socialization isn't just about learning to "play" with others and knowing about popular media. It is about manners, taking turns, knowing how to interact in a variety of situations, knowing how to act with a variety of people, learning about tact . . . If you are getting out and about yourself (with them along), addressing social skills as a part of your curricula and exposing them to others that have a variety of outlooks and experiences, you are socializing them. In terms of taking turns and sharing -they're probably learning that from time with siblings.
Public School Vs Homeschool Socialization By Joan Vasquez Homeschool Provides Better Socialization by Mary Peterson Nuh Uh! Homeschoolers Are Too Socialized! By dreahwrites http://www.tandfonline.com/loi/vtch20 http://www.tandfonline.com/doi/pdf/10.1080/0161956X.2000.9681938 Homeschooling: Breaking The Myths About... by Paula Sloan Socialization Does Occur Within Homeschooling Families by Brenda Hoffman Public School Vs. Homeschool:Who Can Do A Better Job At... by Brenda Hoffman Socialization: Why School Is The Worst Place To Get It by Nan Got Socialization?: A Homeschooling Father's... by Robert Keating Socialization In School?: Is Socialization A Reason For Or Against... by Karen Kaiser http://school.familyeducation.com/home-schooling/human-relations/56224.html http://www.washingtontimes.com/news/2009/dec/13/home-schooling-socialization-not-problem/ www.hslda.ca/cche http://www.homeschool.com/articles/socialization/default.asp http://learninfreedom.org/socialization.html For a good laugh on the subject, check out "Messy Mondays, Seven Lies about Homeschoolers" by Blimy Cow. on You Tube (not full of statistics but true in a sarcastic and funny sort of way every highschool homeschooler I know loves it)