If you do not mind using a 3rd party library, you can download Spire.Doc (also available on NuGet) and referce the DLLs in your project, and use the following C# code snippet to donwload a Word document from a URL.
using Spire.Doc;
using System.IO;
using System.Net;
namespace DownloadfromURL
{
class Program
{
static void Main(string[] args)
{
Document doc = new Document();
WebClient webClient = new WebClient();
using (MemoryStream ms = new MemoryStream(webClient.DownloadData("document's url")))
{
doc.LoadFromStream(ms, FileFormat.Docx);
}
doc.SaveToFile("result.docx", FileFormat.Docx);
}
}
}