The diplomatic and consular missions in the United States cannot issue or renew Belgian driver's licenses.
In Belgium, only residents (both Belgian and non Belgian) who are registered with a Belgian local authority (commune/gemeente) can obtain a Belgian driver's license. In accordance with the European regulations, European driver's licenses are always issued by the local authorities in all European member states.
This is also the case in the United States where the state authorities issue driver's licenses.
If you are a Belgian citizen residing in the United States, you should obtain an American driver's license. In some states, you may be able to exchange your Belgian driver's license for an American one; in other states, you have to pass a test. Please contact the Department of Motor Vehicles of your state for specific information.
It is perfectly legal for a Belgian citizen who resides in the United States to drive in Belgium and europe with a valid U.S. driver's license when visiting as a tourist or for business. To drive in the U.S. a Belgian citizen must have a valid driver's license. Some states require an International Driving Permit (IDP) from foreign nationals, in addition to a valid license from their own country. Check with the motor vehicle department of each state you will drive in for its requirements. (Link: https://www.usa.gov/motor-vehicle-services)
Once a Belgian citizen returns to Belgium on a permanent basis and establishes residence in Belgium, (s)he should once again obtain a Belgian driver's license from the "commune/ gemeente". In most cases, the U.S. driver's license can then be exchanged for a Belgian one.