Is this how the name Pepsi was formed?
No. This is an urban legend.
The name Pepsi-Cola has been around since about 1898.
Cola is a term based on the African kola nut and it was used for its caffeine content. Caleb's drink didn't contain either the kola nut or any caffeine, but it did taste pretty close to the already popular "Coca-Cola" and that was the reason for using the term "Cola" in its name. The "Pepsi" part of the Pepsi-Cola name comes from pepsin, an enzyme which aids in digestion and was also a popular ingredient in early soft drinks (and chewing gum). There has been some dispute as to whether or not the original Pepsi-Cola actually contained pepsin as an ingredient.
The term "Pepsi" in its name is surely an indicator. One of Pepsi-Cola's earliest known advertisement is found in the Feb. 25th, 1903 New Bern Daily Journal, and one of it's claims was that it "Aids Digestion" -- a popular claim for items containing pepsin. Lastly, another newspaper ad produced in 1908 flatout said "PEPSI-Cola is an absolutely pure combination of pepsin -- that's what your stomach needs these days -- acid phosphate and the juices of fresh fruits." (However, it is a fact that by 1923 Pepsi-Cola no longer contained pepsin as an ingredient).
This explains two things that contradict the original claim:
Pepsi-cola is named after Pepsin, not a political slogan.
Even if it wasn't named after Pepsin, it was named over 50 years before the formation of the State of Israel, meaning the slogan would have been meaningless at the time.
[Source: Soda Museum]
A picture of a bottle of Pepsi-Cola from the 1930s, showing the brand pre-dates Israel.