I think there are two questions here.
Do the test results correlate to English skills?
Do the test results correlate to academic success?
Here is a quote from an article on the subject at The Linguist Library, talking about one such test:
ETS has published studies showing a close correlation between the TOEIC test results and results from alternative ways of evaluating English speaking skills, such as interviews. There are other independent studies which resulted in a lower level of correlation between TOEIC results and results from other measurements of language skills. Obviously any evaluation of language skills cannot be absolutely accurate. In comparing the results of two different language tests, at best we can measure the correlation between two imprecise evaluations.
That shows some differing opinions on the first question (see the page for references).
ETS claims that TOEIC is designed to be a good predictor of writing and speaking skills, even though those are not tested in TOEIC. Perhaps this is so. However, even with TOEFL, which does measure writing and speaking skills, people with similar scores will often have quite different abilities to speak and write English. There are criticisms that some students with high TOEFL scores cannot communicate well in English and do poorly at English speaking universities. Similarly, studies of IELTS results and academic success show a surprisingly low level of correlation between test scores and academic results. See TL Forum 1999: Dooey.
This addresses the second question.
Should you trust the views of The Linguist Library? I am not familiar with their work, and I can't comment on their rigour.
p.s. I can understand a university declining admittance based on English skills if they could show that poor English test scores correlated very strongly with poor academic performance. However, declining a student the opportunity to graduate, when they have implicitly demonstrated basic communication skills by passing the other requirements of the course seems a bit harder to justify.