Ultimately the DCT calculation is just one step in the whole JPEG encoding process. A complete implementation also has to deal with quantization, Huffman encoding, and conforming with the JPEG standard.
Java effectively just gives you an interface to a JPEG encoder that lets you do useful things like save images.
The ImageWriter that ImageIO.write() uses for JPEG images depends on your system. The default ImageWriter for JPEGs will only let you change some settings that affect the quantization and encoding using the JPEGImageWriteParam class (http://docs.oracle.com/javase/6/docs/api/javax/imageio/ImageWriteParam.html).
Getting your hand-crafted DCT coefficients into a JPEG file could potentially involve writing an entire JPEG library. If you don't want to do all that work, then you could modify the source of an existing library so that it uses your DCT coefficients.