No, you can't. By putting the image in the CSS it becomes part of the presentational aspect of the web page and hence it is no longer part of the content (separation of form and content).
When someone with a visual handicap visits your site with a text-only or text-to-speech browser the image 'under' your H2's is effectively not there for them, hence you don't even need to add an "alt" text.
Note: "alt" text is supposed to give a description of visual elements in the content of the page, however if you want to create a "mouse tip" balloon text when you hover over a certain element, you should always use "title", which can be added to virtually any tag you'd like.
So, this is possible:
<h2 title="some mouse tip text goes here">h2 Title goes here</h2>