Unit Testing a Public method with private supporting methods inside of it?

When trying to perform test driven development on my JSF app, I have a hard time understanding how to make my classes more testable and decoupled.. For instance:

@Test
public void testViewDocumentReturnsServletPath(){
DocumentDO doc = new DocumentDO();
doc.setID(7L);
doc.setType(“PDF”);
DocumentHandler dh = new DocumentHandler(doc);
String servletPath = dh.viewDocument();
assertTrue(servletPath, contains(“../../pdf?path=“);
}

This is only testable (with my current knowledge) if I remove some of the supporting private methods inside viewDocument() that are meant to interact with external resources like the DB.

How can I unit test the public API with these supporting private methods inside as well?