No.
First of all, sign languages are independent, complete languages, and are not dependent on spoken languages. For example, ASL (American Sign Language) is completely independent from English, so it is possible for a person to be monolingual in ASL. In other words, it is possible that ASL could be the only language that a person knows how to use. However, many deaf people learn to read English since it opens up access to so many documents for them, but they use it more or less as if it were a foreign language. This is why many documents are translated from English to ASL. Sign langauges are just as complex as spoken languages and can communicate the same kinds of things.
That said, the reason there are different sign languages in different parts of the world is the same reason that there are different spoken languages in different parts of the world: new languages are constantly being created. Wherever people want to communicate with each other, they may use gestural information to communicate. A very good example is the creation of Nicaraguan Sign Language, which has only happened in the past 30 years or so. During the Sandanista government in Nicaragua, all the deaf children were placed into the same school together. Previously these children had communicated with their families through gestures, but different families had different gestures. When these kids were put all together in the same institution, they found it necessary to create a new language to communicate with each other. Beginning with the gestures they previously used, the children soon developed a complete signing system. Now new children in the same school learn this language from their older classmates. Voila! Nicaraguan Sign Language was born, and it was never based on a spoken language.
A quick google search should turn up a list of sign languages.