Trying to build a web service that approves the user.  Basically something that would duplicate the functionality of the:

 Manage Accounts >   Authorize User button 

 Does anyone have examples of that? 

Something like this might point you in the right direction...

public HttpResponseMessage AuthorizeUser([FromBody] string username)
{
	var user = DotNetNuke.Entities.Users.UserController.GetUserByName(PortalSettings.PortalId, username);

	if (user != null)
	{
		//UserController.ApproveUser(user);
		user.Membership.Approved = true;
		UserController.UpdateUser(PortalSettings.PortalId, user);

		//Update User Roles if needed
		if (!user.IsSuperUser && User.IsInRole("Unverified Users") && PortalSettings.UserRegistration == (int)DotNetNuke.Common.Globals.PortalRegistrationType.VerifiedRegistration)
		{
			UserController.ApproveUser(user);
		}
	}

	return Request.CreateResponse(HttpStatusCode.OK, "Success");
}

This is based loosely on the code from the page you mention: (Source Root)\Website\DesktopModules\Admin\Security\Membership.ascx.cs