I found the solution! The solution was setting the header from the controller method.

I will share it for anyone who may need it.

public class MenuController : UmbracoApiController
{

    [HttpGet]
    public HttpResponseMessage GetJsonNavigation()
    {
        var linkItems = new List<LinkItem>();

        var homepage = Umbraco.ContentAtRoot().First();
        var menuItems = homepage.Children.Where("Visible");

       linkItems.Add(new LinkItem{Id = homepage.Id, Name = homepage.Name, ParentId = 0});
        foreach (var item in menuItems)
        {

            if (item.DocumentTypeAlias == "Shortcutlink")
            {
                // todo
            }
            else
            {
                linkItems.Add(new LinkItem { Id = item.Id, Name = item.Name, ParentId = item.ParentId });
            }

            var subMenuItems = item.Children.Where("Visible");
            foreach (var sub in subMenuItems)
            {
                if (sub.DocumentTypeAlias == "Shortcutlink")
                {
                    // todo
                }
                else
                {
                    linkItems.Add(new LinkItem { Id = sub.Id, Name = sub.Name, ParentId = sub.ParentId });
                }
            }

        }

        var response = Request.CreateResponse(HttpStatusCode.OK, linkItems);
        response.Headers.Add("Access-Control-Allow-Origin", "*");

        return response;
    }