No you cannot do that because a web part doesn't present a URL. You need to build a webservice.
You can find enough information on the internet (a web part is in ASP.NET a User Control).
Here the same question on Stackoverflow :
how to use jQuery ajax with asp.net user controls?.
You can also have an look at this article on DevNet where they explain how to this with the smart search web part, it is similar to your solution:
Predictive smart search without postbacks
Good luck with coding!!!!
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