Well I guess we should first establish that the "holiness" of the Nile really was the Inundation of it. Hapy, the God that you mentioned, really does represent that facet of the Nile more than anything.

In that way, it was a lifegiver to the people. It seems that that was its primary importance to the Egyptians.

During the reign of Hatshepsut, some small bit of attention was given to the Goddess Iat at Deir el Bahri. Later on in Thutmose III's reign, a temple may also have been built to honor Sobek at another location. Both had some attachment to the Nile's life giving/stealing power, but weren't too popular of Gods. On the contrary, in the same time period, we see a tremendous amount of innovation and creativity go into temples to Amon and other superior deities.

With that in mind, other rivers would obviously not have that kind of impact on them.

So when the Egyptians travelled up to the Euphrates river under Thutmose III, they rejoiced at the sheer adventure of the whole thing. Many of those in his administration commemorated the event in their tombs. Everyone wanted to have crossed the Euphrates.

But in terms of the river itself, well they called it "the river that flows backwards" and that was that.

(2 great sources on this time period include "Thutmose III: a New Biography" by Cline and O'Connor, and "the Wars of Syria and Palestine" by Redford.)