SQL integrated security using computer name instead of user name

Your web service is running under the NT AUTHORITY\Network Service security context. This will cause the process to use the host's machine account when accessing network resources in the domain.

You'll have to configure the web service to run with your domain identity.

If you're hosting your web service in IIS, you can do this through impersonation. Here's an example:

<configuration>
    <system.web>
        <identity
            impersonate="true"
            userName="ROMANIA\username" 
            password="********" />
    </system.web>
</configuration>

That is because your web server is not set up to use the identity of the person using the service, but rather the identity of the computer it is running on.

Check if your web.config for the web service contains: <authentication mode="Windows" />