Rails routes: GET without param :id

The way to go is to use singular resources:

So, instead of resources use resource:

Sometimes, you have a resource that clients always look up without referencing an ID. For example, you would like /profile to always show the profile of the currently logged in user. In this case, you can use a singular resource to map /profile (rather than /profile/:id) to the show action […]

So, in your case:

resource :user do
  get :me, on: :member
end

# => me_api_user GET    /api/users/me(.:format)            api/v1/users#me {:format=>"json"}