。
I can't just silently overlook that problem.
Literal
I [topic-は] that problem [object-を] silently overlook cannot-be-a-case-that.
~わけにはいかない is a strong idiomatic construction meaning 'cannot (for reasons of social/moral obligation)' — not a simple inability, but 'it wouldn't be appropriate.' 見過ごす combines 見る + 過ごす ('to look + to pass/spend') = 'to let pass, to overlook.' 黙って is the te-form of 黙る ('to be silent') used adverbially.