No you cannot. GRE is mandatory for applying to the Fulbright program even if you are applying for an MBA degree or another degree that requires the GMAT. All applicants must plan to take the general GRE and if selected and if required they may be asked to take the GMAT.
Although the GRE is not much different from GMAT in terms of the content, the later tends to be a more difficult exam. Therefore I would say that in a way, it is a good thing that you have to take the GRE. However I am against taking both the tests before the Fulbright deadline as it would add to the costs of taking the test and also to the amount of work one has to do. It is better to take the GRE first before the deadline and then take the GMAT so that you have ample time to take the test at ease.
To register for the GRE please go to the ETS website.
To register for the GMAT please go to the GMAC website.