No, you don't, but it's very very helpful. If you don't have a computer to sync with, your iPad will be an island: the only data that's on there is data you download directly onto it. The only apps you'll have are the ones you've bought from the App Store. Same thing for music/movies/video, except those would come from iTunes on the device.
Having a computer to sync with means you have a way to store more music, apps, videos, movies etc. than you can hold on your iPad, and you have someplace where you've got that data backed up, in case you somehow brick your iPad.