Yes, most state laws allow the licensing authority to suspend your driver's license if were driving without at least the state required insurance coverages, or other approved financial responsibility coverages for your motor vehicle. The other person can come after you for the damages you caused them and if you cannot pay or agree to a payment plan then the person can take you to court and seek a judgment against you.
Even if the other party was insured and used their own insurance for their damages you can still be sought after for reimbursement. Most insurance companies will subrogate with an at-fault party and thus want repayment for the monies they spent on a claim that was not their insured's fault.
If you cannot pay or agree on a payment plan with the other driver or their insurer, whoever is requesting the money for the damages, then most states will suspend your driver's license as a way to try and get you to take responsibility and pay for your actions. Some states may reinstate your license after a certain amount of time has passed while other states will suspend your license indefinitely meaning that until you pay or make approved arraignments to pay they will not reinstate your driving privileges.
For example in Florida their Department of Highway Safety and Motor Vehicles (HSMV) states that if you were involved in a crash with property damage only, no bodily injuries, where you were at fault and were uninsured you should immediately purchase an insurance policy with at least the state required minimum coverages.
Next you should negotiate with each person that sustained property damages to obtain releases for their damages. If the person is insured, you should negotiate with the person's insurance company as the company assumes the rights of compensation (thru subrogation) since they have reimbursed the insured for their loss. You can obtain a release form from your nearest driver license office.
If you are unable to pay the amount of damages in full, you may pursue a monthly payment agreement. With the insurance coverage and releases or monthly repayment agreements, you will be able to provide them upon notice from the department to avoid unnecessary suspensions of your driver license, tags, and registrations. Thus in Florida you can lose your license as well as car registration and license plates until you have paid or set up a repayment plan.
Another example is Rhode Island. Here the law states that the Division of Motor Vehicles can immediately suspend the license and registration and any nonresident's operating privilege, of any person against whom a judgment for payment is rendered against.
In RI the license, registration, and nonresident's operating privilege shall remain so suspended and shall not be renewed, nor shall the license or registration be thereafter issued in the name of the person, including any person not previously licensed, unless and until the judgment is stayed, satisfied in full, or in approved installments, and until the person gives proof of financial responsibility.
While state laws differ it is highly likely that if you do not at least set up a payment plan with the other party that indeed your license will be suspended since the other person needs to be compensated for their damages they sustained by you being at-fault in the accident while driving without insurance. This is one main reason it is important to not have lapses in insurance coverage and to make sure when you do have insurance coverages that your limits are high enough to cover you and your assets in case you are sued in court.
To find out about your state laws you can contact a lawyer that works in this area of law. You may also try contacting your state's insurance regulator and/or Department of Motor vehicles. Either state agency should be able to advise you on the penalties the state can place on you for driving without insurance and if you are unable to pay on the damages you caused to another party.