Most UDDI registry implementations have been built with Java, so that's empirical proof that you can create your own UDDI registry using Java APIs.
But building a UDDI registry is a very large project. Rather than building your own registry from scratch, I suggest you start with an open source implementation such as jUDDI (see http://ws.apache.org/juddi/).
A UDDI service is a Web service that requires a transactional data store. So in addition to Java APIs, you will also need a Web services platform, an HTTP server, and a database (typically a relational database). The UDDI spec defines a set of SOAP APIs (described with WSDL), and it defines the expected behavior of the registry for each API operation. You will have to write the business logic that processes the SOAP APIs and implements the expected behavior.
But perhaps you meant to ask whether you can access a UDDI registry using Java APIs. The answer to that question is also yes. You can use JAX-RPC to access a UDDI registry. As I said, a UDDI registry is a Web service. Its SOAP API is defined with WSDL. Simply compile the WSDL using JAX-RPC and go. But again, it's easier to start with a client library. I suggest using UDDI4J (see http://www.uddi4j.org), an open source UDDI client library.
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