Can I print the Print Date or Save Date in Google Documents?

Yes, it is possible. However, it is not something that is really possible utilizing the existing API. Really the lack of fields and the associated capabilities like in Word is a major disability for document automation.

Something like this will insert a footer on document open and reopen, but it is ultimately not a good solution.

function onOpen() {
    var time = new Date();
    time = Utilities.formatDate(time, "GMT", "yyyy-MM-dd'T'HH:mm:ss'Z'");
    var doc = DocumentApp.getActiveDocument();
    var s;

    if (!doc.getFooter()) {
        var footer = doc.addFooter();
        var footerText = footer.appendParagrap(time);
    } else {
        var footer = doc.getFooter();
        var footerElements = footer.getNumChildren();
        var para = footer.getParagraphs();
        s = para(footerElements-1).getText();
        var footerText = footer.replaceText(s , time);

EDIT: How to add the script.

This code as I have presented above really should be in its own function called from the OnOpen() function rather than in there wholesale, but all good for example purposes.

If you go the Tools menu of an open document then down to Script Manager, when it loads check that it is currently empty, if has a records there, you are on your own for placement of the code.

Close the script manager, back to the Tools Menu and select Script Editor. Now if you had nothing in your script manager, select all the code and delete it. You can then paste in the code provided above and then save it.

Close the document and re-open and you should find there is now a Time Date in the footer. It should update on every open of the document, nothing fancy but will get you going.

Now for those of you who have code already associated with your document. Adding the code above minus the first and last lines into the onOpen() function will also work, but it may not due to code already present conflicting.

Hope this gets you going.