There are numerous inaccuracies in this piece that make the predictions very unlikely. Firstly, the estimate of 5% of patients requiring critical care is high — this has not been echoed in all other places in the world.
More importantly, the figure for number of ICU days used is not referenced and, as far as I can tell, very far off. Most studies indicate a median ICU length of stay of ~7–10 days, although of course this is challenging to perfectly ascertain.
The exponential model also does not take into account individual reactions to the COVID-19 outbreak. While it is possible that society will in no way change as the number of cases mounts, we are already seeing people take significant steps towards social distance, as well as government intervention. It is a very unlikely assumption that we will see the exponential curve continue for the better part of a month unchecked.
The description of what R² implies is also not correct. This does not describe the confidence in the model, it is the proportion of the increase in cases explained by the model. Given that the model was created largely using this data, it is inevitable that the R² would be high, and does not speak to the confidence in the model in any way.
There are other mistakes, but these explain why the model is extremely unlikely to accurately predict the situation with ICU beds in Australia.