From the tests done I get this error: java.lang.IllegalStateException: getWriter () has Already Been Called for this response.
In other words, you were trying to mix the binary data of the image with the character data of the HTML output, or you were trying to do this in a JSP instead of a Servlet. This is indeed not going to work. You need to send either the image or the HTML page exclusively in response to fully separate requests.
In your JSP/HTML page just have a link to the image, like so:
<a href="imageservlet/filename.gif">click to download image</a>
Then, in a servlet listening on an url-pattern of /imageservlet/*, you just get the image as InputStream from some datasource (e.g. from local disk file system as FileInputStream) and then write it to the OutputStream of the response the usual Java IO way.
You only need to set at least the Content-Disposition response header to attachment to make sure that the client get a Save As popup dialogue, else it will be displayed straight in the browser. Setting the Content-Type and Content-Length are also important so that the browser knows what the server is sending and can predict how long the download may take.
response.setHeader("Content-Type", getServletContext().getMimeType(file.getName()));
response.setHeader("Content-Length", String.valueOf(file.length()));
response.setHeader("Content-Disposition", "attachment;filename=\"" + file.getName() + "\"");
You can find complete basic servlet example in this article.
Note: you cannot control where the client would save the image, this would be a security hole. This way websites would be able to write malicious files on client's disk unaskingly.
Update: as per your update, there are two options:
You need to let the client itself fire two HTTP requests (I've answered this in your subsequent question)
Create a client side application which does all the task directly at the client side and then embed this in your webpage, for example a Java Applet. With an applet you have full control over the client environment. You can execute almost all Java code you'd like to execute and you can write files to disk directly without asking client for the location to save. You only need to sign the applet by a 3rd party company or the client needs to confirm a security warning before running.