No. From A-to-z Claims Process for Property Damage and Personal Injury:
For customer claims of $1000 or less, Amazon will resolve the claim through a concession to the customer and will not request reimbursement from you or your insurer if you provide proof of insurance. Upon receiving a claim of $1000 or less, Amazon or our third-party administrator will attempt to resolve the claim directly with the customer while providing you with details about the claim. In addition, you will have an opportunity to respond to the customer or to us with any information you feel is relevant to our review of the claim, such as evidence that your product was not defective or did not cause the customer’s damage or injury. If we are able to resolve the claim with the customer, we will ask you to provide proof of your insurance within 7 days (if you have not already done so through the Business Insurance page). If you already have insurance or if we request that you obtain it and you subsequently provide proof of coverage, we will not request reimbursement from you or your insurance provider for the amount we pay to the customer as a concession for their claim.