I'll assume your product is electrically powered. Third party certifications are available and relevant in other contexts, but electrical conformance is by far the most common.
There are two ways that a third party certification can be important:
1. The FDA as a part of the pre market clearance process...510(k) or PMA...may require evidence that your device conforms to relevant standards. For devices that use electrical power, the FDA usually considers IEC 60601-1 relevant, and other standards may be applicable as well. One way to efficiently show the FDA that you have such conformance is to have certification from a third party lab.
2. When you market your device, particularly if you sell directly to hospitals, you may find that buyer-expectations for certification are important. Hospitals often have biomedical engineering personnel, architects, risk managers, legal staff and others who impose purchase specifications to control liability exposure and to conform to Joint Commission and regulatory expectations. In such a marketing context, a third party certification...traditionally called Listing or Labeling, and indicated with a brand mark or label on the device near its electrical connection point...can be important.
This consideration is elevated significantly when a device is permanently wired to the electrical supply, rather than being cord-and-plug connected. In many US jurisdictions, the local electrical code enforcement authority will not permit installation of equipment that is not Listed or Labeled. That stance is consistent with the most commonly used electrical construction/installation codes in the US.
Note by the way that UL is the oldest, but by no means the only, Nationally Recognized Test Laboratory (NRTL). In the US, the Occupational Safety and Health Adminstration (OSHA) supervises the NRTL program. Local electrical code enforcers should accept a certification from any authorized-within-scope NRTL on an equal footing. A list of all the NRTLs is available at the OSHA site.