I hate the D**** GMail app. It does everything exactly WRONG.
I log in to my GMail account using Thunderbird on my PC. All messages are downloaded. Many are deleted. But my Android phone continues to waste my money and space downloading the same messages.
Not only does the Android GMail App insists on repeating messages that Thunderbird has wiped, but deleting a message on Android GMail results in Thunderbird never seeing that message at all.
Ok, you have to decide: do you want Thunderbird to see every message, or do you want GMail and Thunderbird to stay in sync?
It sounds like you area accessing your account using POP rather than IMAP on Thunderbird. If you want to synchronize your mail, so that anything that you read on TBird is marked as read on GMail, and anything that you delete on TBird does not appear in GMail on Android, then you are also gong to have to live with the fact that if you delete a message in Gmail, it will be deleted in Thunderbird as well.
If this is the case, then you can continue to use POP on Thunderbird, and that's fine, but that makes all of your mail download to Thunderbird and any actions that you take on it - reading, deleting - will not be communicated back to Gmail, so the GMail app on Android will look to be downloading duplicate messages. However, if you want everything to stay in sync, you first need to enable IMAP access in Gmail on the web (go into a web browser, attach to your Gmail account, click the "Gear" icon on the top right, select "Settings", go to "Forwarding and POP/IMAP", and make sure that IMAP is enabled. Then you will have to set up your Gmail account in Thunderbird as an IMAP rather than POP connection (I would create a new account - you can always delete the POP account later, but if you do not like IMAP, you can easily go back to the POP account if you create a new one.)
Now, anything that you do in Thunderbird - read mail, reply to it, delete it - will be synchronized with Gmail on the web and, hence, GMail in Android.
Android GMail is designed to be your primary, your only, e-mail client. For me it is not; it is a seconary client which could be useful when I'm not near my PC. And Android GMail does not have the options to make it a decent secondary E-mail tool. I hate it so much I'd like to kill it. Anyone know what the Linux binary is named?
If you don't want to delete it but simply want to stop using it, you can always go into (from home) settings->accounts & sync->your gmail account ad uncheck Gmail from syncing options - this will stop Gmail delivery to your phone (unless you go into GMail and hit menu->refresh.)
If you are rooted and really want to stop it, the gmail.apk file is in /system/apps/gmail.apk. I'd rename it with Root Explorer before deleting it, in case the phone crashes without the app.
Let me also warn you that if you are not using GMail on the phone, the stock mail client has an issue that it never releases storage space taken up by messages even if you delete them and empyt the trash. I recommend using the free app K-9 mail in the market instead.