No. Insurance or a license is not a factor as to who's at fault. However, the other driver's insurance may contact you asking to pay for damages.
More importantly, your husband shouldn't have been driving with a suspended license and no insurance. What would you have done if he clearly was at fault and he seriously injured or killed someone?
Daniel is correct. His lack of a drivers license is not relevant whether or not he was negligent. However, it is relevant whether or not you were negligent in allowing him to use your vehicle if you knew that he was unlicensed. We call that negligent entrustment. If you are both sued, insurance company should defend you, but might refuse to defend your husband. If you're served with a lawsuit, you should turn it into the insurance company, demanding defense and indemnity.
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Lack of a drivers license does not affect who is at fault.
None of the opinions above should be considered legal advice, until I have had the opportunity to review the actual facts of the case.
Well stated by Daniel Horowitz and John Zgourides. The mere fact that your husband had no license doesn't automatically put him at fault, but this does not sound like a clear liability accident. If the other driver has to file on her own insurance for her repairs, her insurance company will likely look to yours for reimbursement. Your insurance company almost certainly won't pay since your husband was excluded from coverage. At that point the other driver's insurance company may report your husband to the DPS for driving without insurance, which usually results in a driver's license suspension, but your husband has no license to suspend. The other driver's insurance company may also sue your husband to try and recover the money they paid out for the other driver's repairs.
Your husband needs to get his driver's license in order and get some insurance coverage. Imagine how angry you or your husband would be if an uninsured driver totaled your car and injured one or both of you.
Disclaimer: answers posted by attorney Daragh Carter to questions posted on AVVO are NOT privileged or confidential and will not and should not be construed to create an attorney-client relationship between attorney Daragh Carter and you or anyone else.
Insurance and fault are separate issues. Not having insurance, while problematic for other reasons, will not affect whether or not your husband was at fault.