In The Earth is Precious, Chief Seattle says that man did not "weave the web of life, he is merely a strand in it." What attitude is he implicitly criticizing by saying this?

By the sentence given as an example, the attitude that is being criticized by Chief Seattle in The Earth is Precious is the attitude of not taking control of what happens in life.

When saying "wave the web" he is referring to create one of his own, not being part of an existing web made probably by others, like a "strand".

By taking control of your life, or anything in our lives, we cannot blame anyone but ourselves for what happens, we take responsibility for what happens and we are also capable of changing the way we are weaving any time we want.