While I believe moral absolutes are necessary, I don't think that the issue this post is posing is one that stems directly from the Common Core Standards. I have heard of problems with CCS, but most issues I hear of it are one of two camps:
(a) people that are uneducated in what CC is about and instead rely on others to tell them what the standards mean or imply or
(b) people familiar with CC that fail to understand the purpose behind it.
Don't get me wrong, I do feel there are things to iron out from CC, but these issues arise most frequently at the elementary level.
The ultimate goal of CC is (at least where math is concerned and where I believe Science is headed when it is completed for the most part) is to grant more freedom to the educator to teach concepts and get students to dig deeper into Bloom's taxonomy where they analyze/evaluate/create--meaning "critical thinking" as opposed to being more rigid like Quality Core that tends to direct learners to the lower end of Bloom's toward remembering/understanding which relies more on rote memorization and short-term memory retrieval.
No offense toward elementary school teachers, but most of them don't have the ability to get young children thinking critically, and merely focus on memorization. Some do, and it's seen in exceptional students at the higher levels.
When teachers have students memorizing information with low - application based on such memorizing, then the results are what the article is showing: students that cannot see where belief and fact coexist, or the existence of moral absolutes - such as in the case of laws, codes of conduct, etc.
Where the article is correct is in this facet--students are asked (at young ages, unfortunately) to base an answer to fact/opinion by verbage rather than by justification via evidence or counter-example.
Educationally, there has been a decline in morality views and the increase in "gray area truths" like cheating, fighting, theft, dishonesty, etc. and they have been increasing prior to the induction of CC. It just happens to be that CC doesn't quell these ideas.