If the the original sentence is for example,
アイツ帰るなり、すぐに部屋に 横になった。
As soon as he came home, he laid down in his room.
If you omit verb 帰る,
アイツなり、すぐに部屋に横になった。
アイツなりdoesn't make sense, and cannot translate.
If you omit verb 横になった、
アイツ帰るなり、すくに部屋に。
he did what??? after he came home?
There is no information what you want to say in this sentence.
so you don't omit verb the sentence above.
However ,we sometimes omit verb for the next situation.
彼は学校でピアノを練習した。そして、家に帰ってからも練習した。
He practiced the piano at school. He also practiced the piano after he came home.
you can omit the verb 練習した in the second sentence because you don't have to say 'practiced ' again.
彼は学校でピアノを練習した。そして、家に帰ってからも。