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Код:

unit Unit1;

interface

uses
  Wininet,
  Windows, Messages, SysUtils, Variants, Classes, Graphics, Controls, Forms,
  Dialogs;

type
  TForm1 = class(TForm)
    procedure FormCreate(Sender: TObject);
  private
    { Private declarations }
  public
    { Public declarations }
  end;

var
  Form1: TForm1;

implementation

{$R *.dfm}

function GetInetFile(const fileURL, FileName: string): boolean;
const
  BufferSize = 1024;
var
  hSession, hURL: HInternet;
  Buffer: array[1..BufferSize] of Byte;
  BufferLen: DWORD;
  f: file;
  sAppName: string;
begin
  Result:=False;
  sAppName := ExtractFileName(Application.ExeName);
  hSession := InternetOpen(PChar(sAppName),
  INTERNET_OPEN_TYPE_PRECONFIG,
  nil, nil, 0);
  try
    hURL := InternetOpenURL(hSession,
    PChar(fileURL),
    nil,0,0,0);
    try
      AssignFile(f, FileName);
      Rewrite(f,1);
      repeat
        InternetReadFile(hURL, @Buffer,
        SizeOf(Buffer), BufferLen);
        BlockWrite(f, Buffer, BufferLen)
      until
        BufferLen = 0;
      CloseFile(f);
      Result:=True;
    finally
      InternetCloseHandle(hURL)
    end;
  finally
    InternetCloseHandle(hSession)
  end;
end;

procedure TForm1.FormCreate(Sender: TObject);
begin
  GetInetFile('http://www.delphisources.ru/pages/faq/base/get_file_from_net.html', 'd:\Downloads\get_file_from_net.html');
end;

end.