Uses WinInet;

function GetHttpString(URL: AnsiString; var OutData: AnsiString): Boolean;
var
 hInet, hFile: HINTERNET;
 DataBuf: Pointer;
 TmpStr: AnsiString;
 BytesReaded: Cardinal;
const
 BufSize = 4096;
begin
 result := false;

 hInet := InternetOpen(nil,
   INTERNET_OPEN_TYPE_PRECONFIG,
   nil,
   nil,
   0);
 if hInet = nil then Exit;

 hFile := InternetOpenUrl(hInet,
   PChar(URL),
   nil,
   0,
   INTERNET_FLAG_EXISTING_CONNECT or INTERNET_FLAG_NO_CACHE_WRITE or INTERNET_FLAG_RELOAD,
   0);
 if hFile = nil then
 begin
   InternetCloseHandle(hInet);
   Exit;
 end;
 OutData := '';
 GetMem(DataBuf, BufSize);
 BytesReaded := 0;
 repeat
   SetString(TmpStr, PChar(DataBuf), BytesReaded);
   OutData := OutData + TmpStr;
   BytesReaded := 0;
   if not InternetReadFile(hFile, DataBuf, BufSize, BytesReaded) then Break;
 until BytesReaded = 0;
 FreeMem(DataBuf, BufSize);
 InternetCloseHandle(hFile);
 InternetCloseHandle(hInet);
 result := true;
end;

...
var
s: string;
fs: TFileStream;
begin
s := IdHTTP.Get('http://www.url/'); // Параметры запроса!
fs := TFileStream.Create('Get.txt', fmCreate or fmOpenWrite);
fs.Write(Pointer(s)^, Length(s));
fs.Free;
end;
...