As a US citizen, you need to get no objection letter from US Embassay to get marriage certificate from CDO office. One of my friend did not obtain the certificate before getting married and the CDO office gave him hard time to provide with marriage certificate. See more from the Embassay website:

For a marriage in Nepal to be legally-binding, the couple (even if a U.S. citizen) must meet the Government of Nepal's requirements. The authorities responsible for registering marriages involving foreigners are the Chief District Administrative Office for each district, popularly known as "CDO office." In Kathmandu, the CDO office is located in Babar Mahal.

Nepal requires U.S. citizens marrying in Nepal to have an Affidavit of Eligibility to Marry, notarized by a U.S. Embassy Consular Officer and available at the Embassy. This is not a requirement of the United States. There is a USD $50 fee associated with this notarial service, payable in U.S. Dollars or the equivalent in Nepalese rupees. This notarial service can be completed on Monday from 9:00-11:30 am or Friday from 1:30-3:45 pm at the Consular Section at the U.S. Embassy on Maharajgunj, Kathmandu. It is necessary to schedule an online appointment for this service. A copy of the Nepali citizen's passport or citizenship card is required.

Registering your marriage at the CDO

To apply for a Government of Nepal-issued marriage certificate, you must submit the Affidavit of Eligibility to Marry with other required papers at the CDO office. The CDO also requires evidence of your nationality (U.S. passport), photographs, and a nominal processing fee. After you apply, the CDO will check your legal status in Nepal with various government offices. This process may take some weeks. After the CDO processes your application, they will issue a marriage certificate, which is usually written in both Nepali and English.