Which sentence is correct?
My project is dependent upon your project completing.
My project is dependent on your project completing.
They are equivalent; upon is just a more formal term for on, and it is especially used in abstract senses.
The NOAD reports the following note about the usage of upon:
The preposition upon has the same core meaning as the preposition on. Upon is sometimes more formal than on, however, and is preferred in the phrases once upon a time and upon my word, and in uses such as row upon row of seats and Christmas is almost upon us.