I've attempted this with java 6.  
It appears to work correctly. I've succesfully read headers and body content from a file that had a wildcard SSL certificate.
package com.example.test;
import java.io.DataInputStream;
import java.io.IOException;
import java.net.MalformedURLException;
import java.net.URL;
import java.net.URLConnection;
import java.util.Iterator;
import java.util.List;
import java.util.Map;
import java.util.Map.Entry;
public class SSLTEST {
    public static void main(String[] args) {
        try {
            URL url = new URL("https://test.example.com/robots.txt");
            URLConnection connection = null;
            try {
                connection = url.openConnection();
            } catch (IOException e) {
                e.printStackTrace();
            }
            Map<String, List<String>> fields = connection.getHeaderFields();
            Iterator<Entry<String, List<String>>> headerIterator = fields.entrySet().iterator();
            System.out.println("HEADERS");
            System.out.println("-------------------------------");
            while (headerIterator.hasNext()){
                Entry<String, List<String>> header = headerIterator.next();
                System.out.println(header.getKey()+" :");
                Iterator<String> valueIterator = header.getValue().iterator();
                while (valueIterator.hasNext()){
                    System.out.println("\t"+valueIterator.next());
                }
            }
            String inputLine;
            DataInputStream input = new DataInputStream(connection.getInputStream());
            System.out.println("BODY CONTENT");
            System.out.println("-------------------------------");
            while ((inputLine = input.readLine()) != null) {
                System.out.println(inputLine);
            }
        } catch (MalformedURLException e) {
            System.err.println(e);
        } catch (IOException e) {
            e.printStackTrace();
        }
    }
}
EDIT I've just recieved confirmation that this works on java 1.5