Preventing data modification from outer source in SQLite

Restrict user access

Only the operating system can secure files against access by unauthorized persons. Put the database into a folder, which is only accessible by the current user, and have a separate database for each user.

Encryption

You're already encrypting the passwords, that's good. If you want to encrypt the whole database, you could have a look at the SQLite Encryption Extension.

The SQLite Encryption Extension (SEE) is an add-on to the public domain version of SQLite that allows an application to read and write encrypted database files.

Also have a look at the question SQLite with encryption/password protection.

Attack

What would actually happen if someone has access to the database file?

If the database is secured properly, the attacker is not able to get the plain passwords (at least not in reasonable time). In the worst case a password is replaced by another one, but that would achieve nothing, besides you using the wrong password and maybe resetting it. Therefore the worst case would be that you'll lose your saved passwords.

You can do nothing to prevent a data loss on a single machine. For example hard disks sometimes just stop working, someone could steal the whole PC, format the hard disk, etc.

Backups

If you really want to make sure that the data is not modified, you need to keep backups on different machines to minimize the possiblity that someone has access to all of them. For example you could upload the database file to a cloud service. Then you sign the file, so that you can see if a file was compromise and if so fallback to another version.

Conclusion

Your password manager is good enough for an offline tool. If you want to improve the data integrity you have to transfer the data to other machines.