Well, the first thing you need to establish is who was at fault. If you, the bicyclist, were at fault in causing the accident, then no, you cannot sue the driver. However, if the other driver was at fault in causing the accident by acting negligently or carelessly, then certainly you have a claim or a lawsuit that can be brought against them.
You don’t have to proceed right to a lawsuit. If you’ve been injured in a bicycle accident, then you may have a right to bring a claim directly with the at-fault driver’s insurance company and may possibly be able to negotiate a settlement without the need to sue. But, if you cannot come to terms, then you do have the right to sue or bring a lawsuit.