HttpGet

uses
  WinInet;

function HttpGet(const URL: AnsiString): AnsiString;
const
  BufferSize = 1024;
  USER_AGENT = 'Mozilla/5.0 (Windows NT 6.3; WOW64; rv:32.0) Gecko/20100101 Firefox/32.0';
var
  hSession  : HInternet;
  hURL      : HInternet;
  Buffer    : array[0..Pred(BufferSize)] of AnsiChar;
  BufferLen : DWORD;
  dwReserved: DWORD;
  respCode  : DWORD;
begin
  Result := '';
  hSession := InternetOpenA(PAnsiChar(USER_AGENT), INTERNET_OPEN_TYPE_PRECONFIG,  nil, nil, 0);
  try
    hURL := InternetOpenURLA(hSession, PAnsiChar(URL), nil, 0, 0, 0);
    BufferLen := SizeOf(respCode);
    dwReserved := 0;
    if (HttpQueryInfo(hURL, HTTP_QUERY_STATUS_CODE or HTTP_QUERY_FLAG_NUMBER,
                      @respCode, BufferLen, dwReserved)) then
      if respCode = 200 then
      try
        repeat
          ZeroMemory(@Buffer, SizeOf(Buffer));
          InternetReadFile(hURL, @Buffer, SizeOf(Buffer), BufferLen);
          Result := Result + StrPas(Buffer);
        until BufferLen = 0;
      finally
        InternetCloseHandle(hURL);
      end else
        InternetCloseHandle(hURL);
  finally
    InternetCloseHandle(hSession);
  end;
end;

procedure TForm1.Button1Click(Sender: TObject);
begin
  sMemo1.Text := HttpGet('http://my-site.ru/Update/Update_2.0.0.txt');
end;