"He goes house"? This meaning of nach is only really used in front of Hause and country names, I believe, but it's important.

Hi! elae is right. The sentence would not make any sense if you would skip "nach" in front of "Hause". "Hause" would not even have a meaning if it is used without the "nach" before it.

Just remember some fixed combinations: "Er ist zu Hause" = "he is (at) home". "Er geht nach Hause" = "he goes home". The rule is: no article >> nach. With article >> zu. As country names generally have no article elae is right. But: "he goes to the house on the hill" = "er geht zu dem (zum) Haus auf dem Hügel".

For a native german speaker it's as weird as saying "He goes to home" is for a english-native.

I think it is also worthwhile to note here that "zu Hause" and "nach Hause" are special cases. As siebolt said, it would be proper to say "er geht zum Haus auf dem Hügel", but also notice that in this sentence, "Haus" does not have an -e suffix even though it follows "zu" like it does in the expressions "zu Hause" and "nach Hause". There are several nouns in German that get an -e suffix in the dative in certain circumstances, which is a hold-over from different spelling reforms of the past (if memory serves correctly). When you want to express that you are at or going to a home that is explicitly or implicitly owned by someone (Ich bin bei ihr zu Hause), don't forget the -e at the end!