No, the Fourteenth Amendment does not bestow citizenship by birth in the USA, because an embassy is not part of the USA.
As the U.S. Department of State Foreign Affairs Manual Volume 8 notes, US embassies are not part of the United States and do not acquire U.S. citizenship under the 14th Amendment.
8 FAM 301.1-3 NOT INCLUDED IN THE MEANING OF "IN THE UNITED STATES"
c. Birth on U.S. Military Base Outside of the United States or Birth on U.S. Embassy or Consulate Premises Abroad:
(1) Despite widespread popular belief, U.S. military installations abroad and U.S. diplomatic or consular facilities abroad are not part of the United States within the meaning of the 14th Amendment. A child born on the premises of such a facility is not born in the United States and does not acquire U.S. citizenship by reason of birth.
The US State Department notes:
U.S. embassies and consulates abroad, as well as foreign embassies and consulates in the United States, have a special status. While diplomatic spaces remain the territory of the host state, an embassy or consulate represents a sovereign state. International rules do not allow representatives of the host country to enter an embassy without permission --even to put out a fire -- and designate an attack on an embassy as an attack on the country it represents.