Regardless of if you own Half-Life 2, you should still have it's game content. Sounds like the file you're refering to is gameinfo.txt.
Here is the content for gameinfo.txt for Half-Life 2
"GameInfo"
{
game "HALF-LIFE 2"
title "HALF-LIFE'"
type singleplayer_only
FileSystem
{
SteamAppId 220 // This will mount all the GCFs we need (240=CS:S, 220=HL2).
ToolsAppId 211 // Tools will load this (ie: source SDK caches) to get things like materials\debug, materials\editor, etc.
//
// The code that loads this file automatically does a few things here:
//
// 1. For each "Game" search path, it adds a "GameBin" path, in <dir>\bin
// 2. For each "Game" search path, it adds another "Game" path in front of it with _<langage> at the end.
// For example: c:\hl2\cstrike on a french machine would get a c:\hl2\cstrike_french path added to it.
// 3. For the first "Game" search path, it adds a search path called "MOD".
// 4. For the first "Game" search path, it adds a search path called "DEFAULT_WRITE_PATH".
//
//
// Search paths are relative to the base directory, which is where hl2.exe is found.
//
// |gameinfo_path| points at the directory where gameinfo.txt is.
// We always want to mount that directory relative to gameinfo.txt, so
// people can mount stuff in c:\mymod, and the main game resources are in
// someplace like c:\program files\valve\steam\steamapps\half-life 2.
//
SearchPaths
{
Game |gameinfo_path|.
Game hl2
}
}
}
It should go in \steamapps\<username>\half-life 2\hl2\GameInfo.txt
Also if you verified Garry's Mod and had no issues, try validating any other source based games you own, one of them should trigger this content to be valided... It's a bit odd Garry's Mod validation didn't detect any issues though. It most certainly should have.
I would also check the contents of Garry's Mod's gameinfo.txt also, here it is.
"GameInfo"
{
game "Garry's Mod"
title ""
title2 ""
type multiplayer_only
"developer" "Facepunch Studios"
"developer_url" "http://www.garrysmod.com/"
"manual" "http://wiki.garrysmod.com/"
FileSystem
{
SteamAppId 4000 // 218
ToolsAppId 211
SearchPaths
{
Game |gameinfo_path|.
Game hl2
Game |all_source_engine_paths|hl2
}
}
}
It resides in steamapps\<username>\garrysmod\garrysmod\gameinfo.txt