I have successfully updated the page and after publishing the page successfully, still in page view source the page's last published date isn't updating. I have already cleared the cache, still there is the same issue.
It is most probably cache issue on Content Delivery side. To verify that page is correctly published, please check its content on delivery side (on file system or in database). If all is ok, please restart CD apps. It will most probably fix the issue.
Also you should check why page got cached in the first place.