In Terminator 3, John Connor asks the T-850 how Judgement Day could possibly take place after the destruction of the things that fulfilled its preconditions. The T-850 replies, "Judgment Day is inevitable." Why does the T-850 say that?
There are 2 explanations for this:
This is the historical inevitability discussed in Marx's dialectic materialism (you can get more detailed info from Samuel Russell on Politics.SE or History.SE; or may be people on Philosophy.SE).
High level - human scientific and social development is a well-predictable evolution. Slavery, feudalism, industrial revolution, capitalism, social revolution, socialism/communism (the way the sequence was taught in USSR).
Like all progressives, Cameron is more of a Marxist than he either acknowledges or likely even realizes.
So, the same concept underlies the emergence of Skynet and the Judgement day. Socio/political processes all drive towards more emergent technology that is closer and closer to AI capabilities.
Even if there's no T800's arm or a chip to reverse engineer, someone will create the technology from scratch, earlier or later. If not Cyberdyne Systems, then Google or IBM or some dude in a basement in China.
And geopolitics dictates that there will be ample WMDs around for Skynet to use.
One a more fictional level, there's a fictional trope of "Stable time loops".
See Wikipedia or TVTropes for details; but the short version is that HAVING a time loop (e.g. having a Terminator in the past) guarantees that somehow he will be sent in the past, no matter what you do. His presence in the presence is an invariable constant.
And since Skynet and Judgement day are required for T800/T850 to be sent back, they will happen.
It's all about the constants and variables.
There'll always be Judgement day, and there'll always be a protector sent back in time and there'll always be a terminator sent back by SkyNet and John Connor / Sarah Connor are always involved.
Variables though are things like terminator models, the ages of people involved, the date, the attitude of each character, the people who are killed etc.
Edit: The reason Judgement day has to happen is because if it didn't then there'd be no terminator films, no one to send back in time, no need to protect Sarah Connor and no need to terminate John Connor, because John Connor wouldn't exist.