Your API will return something like
return new FileStreamResult(pdfStream, "application/pdf")
where pdfStream is a stream of your PDF, either from a PDF generator that returns the PDF as an in-memory stream, or a stream from a file if it is on the disk.
If you want to retrieve the PDF from this api and show it in the browser you need to read the stream, then re-write the stream to the client.
string url = "http://yourserver/api/getpdf";
WebRequest req = WebRequest.Create(url);
req.Method = "GET";
req.ContentType = "application/x-www-form-urlencoded";
WebResponse response = req.GetResponse();
dataStream = response.GetResponseStream();
Response.Clear();
Response.Buffer = true;
Response.AddHeader("content-disposition", "attachment;filename=test.pdf");
Response.ContentType = "application/pdf";
dataStream.CopyTo(Response.OutputStream);
dataStream.Close();
dataStream.Dispose();
response.Close();
Response.End();
The above is an aspx client page, but the concept for MVC is the same....get the stream from the api and push it to the client response.