"idly by" is a correct and usable expression in written English. It is used to refer to someone or something that is idling away, not doing anything of consequence. For example, "With no classes to attend, she passed the day idly by, lounging in the sun and reading a book.".
We can't stand idly by". .
The West cannot stand idly by.
Congress did worse than sit idly by.
Yanukovych was not sitting idly by.
The Taliban are not standing idly by.
Hot 97 had not sat idly by.
Brussels cannot stand idly by.
I love the desktop app, it’s always running on my Mac. Ludwig is the best English buddy, it answers my 100 queries per day and stays cool.
Cristina Valenza
Retail Lead Linguist @ Apple Inc.