I really feel it's an accurate description. If you get a positive result on the test, there's a 16% chance your fetus has a 1p36 deletion and an 84% chance they don't.
As you said “if you get a positive result”. It’s true, if you ignore the 99.9% of the time the test is correct (true negative result), then you can say the test is 84% wrong.
84% of people who got a positive test result will end up telling their family "it's OK, the first test was wrong, my baby doesn't have a 1p36 deletion after all". The 99.9% of other people who got true negatives are important from a test design perspective, because specificity is closer to the actual levers you can pull on, but it's not super relevant to the decisionmaking process of someone who gets a positive result.
Ignoring all the true and false negatives which themselves are markers of how accurate the test is.
16% precision is the correct statement, saying the test is wrong 84% of the time implies that those getting negative results might actually have positive results.
He framed his statement correctly, limiting his observation to the condition that the test returned a positive result. Saying that 84% of positive results are false is correct if only 16% are true. You'd need to know false negative rates and base occurrence rates (modified by whatever other factors are unique to your situation) to inform the nature of information you get by performing the test.