Pure HTML 5 Alone
No. As the other answers point out, Gmail does not provide a RESTfull API that would be accessible to a HTML mobile app. However, Gmail does provide the standard IMAP and SMTP APIs. You can read all about integrating GMail features within apps here: https://developers.google.com/gmail/.
Pure HTML 5 With Middleware
Using pure mobile HTML5 cross platform code - you will not be able to make use of these APIs. However, you could write a server to act as a middleware between GMail and your pure HTML 5 application. The server could do all the interaction needed with Gmail, and your application could interface with the server. I'm not recommending this because you could be opening up your user's Gmail to many security flaws, and it would be a lot of extra work on top of the HTML5 app - but it is possible.
Mobile Chrome App
Although this is not pure HTML5 - Chrome Apps allow TCP socket connections. Having direct TCP connections would allow you to write - or find an existing - IMAP client to read email. Next, you could use Mobile Chrome Apps to put your chrome app on IOS and Android. This is not pure HTML5 and is not 100% cross platform (windows phone, etc), but you app would be able to directly talk to Gmail.
Platform Specific Apps
Of course the final option is to write platform-specific apps that can directly interface with IMAP & SMTP. This option does go directly against your requirements in the question, but I felt like it should be included to be a comprehensive answer.