US FDA uses two different country of origin statuses.
Labels on devices that cross the US border are expected by FDA to conform to Department of Commerce rules, which are commonly known as the Customs and Border Protection (CBP) Country of Origin requirement. In general, these rules define the Country of Origin as the location of last "substantial transformation". Physical fabrication steps are a substantial transformation. Sterilization is a substantial transformation. Packaging and labeling may or may not be, depending on circumstances. CBP has an extensive set of rules and precedents, and formal procedures for obtaining a judgement as to a product's status.
More recently, FDA began to use the PREDICT system for evaluation of import shipment documentation. PREDICT shipment documentation is expected by FDA to include not the CBP Country of Origin, but instead the FDA Country of Origin, which is defined as the location at which the product legally becomes an FDA Listed medical device.
When a product is made by an offshore fabricator and that company is a 21CFR 807.65(a) component maker rather than a 21CFR 820 device maker, and that offshore company is not Registered, does not List its products as devices, and does not distribute those products to end users, and the initial importer of those products is Registered and Lists those products as devices, then those products legally become medical devices at the US initial importer and that location determines the FDA Country of Origin for PREDICT purposes.
The FDA Country of Origin for PREDICT purposes is not required to be stated on the label, and probably should not be unless the CBP Country of Origin is clearly identified so that no confusion results at time of border inspection.