"critical objective" is correct and usable in written English.
You can use it to refer to a goal with a high level of importance. For example, "The critical objective of the project was to develop a fully functioning prototype by the end of the year."
Since the adequacy of early management of critically ill patients with sepsis including antibiotic administration is thought to be critical, objective markers are required.
Movies for him aren't abstract, cold, critical, objective things.
But it has failed to achieve its own critical objective: sense of place.
It lacked breadth and depth, so expanding the donor pool was a critical objective.
Science can improve our predictions of what might happen; physics can provide critical, objective analysis of new schemes.
Demonstrating to Iran that the Saudi-American alliance remains strong has emerged as a critical objective of the Obama administration.
We have an opportunity to address two critical objectives at once: improving the health and lives of women and children and, simultaneously, helping to slow dangerous climate change.
Minimizing memory requirements for program and data are critical objectives when synthesizing software for embedded DSP applications.
The maps provide a visual representation of a company's critical objectives and the crucial relationships among them that drive organizational performance.
A good annual exercise for managers is to choose their most critical objectives, and then to map the trade show investments that best satisfy these.
In the Philippines, researchers found that the international response to Typhoon Haiyan in 2013, backed up by $1bn in overseas money, had been "timely and met the critical objectives of providing food, water and shelter and preventing significant post-event mortality and morbidity".
