It wasn't published in Nature; it was published in the Proceedings of the National Academy of Science.

The rebuttal was also not published in Nature, but it was closer, published in Nature Reviews Urology.

I think the main basis for the criticism is that they didn't look at clinical outcomes like infertility or ability to conceive. As a result, we only have a (very robust) report of a relationship between ibuprofen and hormone levels, not between ibuprofen and x, where x is some directly important/meaningful clinical outcome.