It's a little dated and brief, but it's a cool article and I believe it's overall sound in it's predictions and arguments presented.
On the overall point, the author isn't predicting an ice age in 2,000 years, so the answer to your title question is that he didn't really say that, though he writes the article somewhat poorly because that's not very clear, especially when he writes the words "double whammy". But he's predicting a natural correction to the added CO2 within about 2,000 years, after which the glaciation cycle, but not necessarily an ice age, should return.
The next ice age is hard to predict because Earth's Milankovich cycles aren't adding up right for another 150,000 years, so if an ice age can be triggered by a small push, one could happen in 10,000 years, but a big push isn't due for some time. More details on the timing of the next glaciation here.
Your author's primary point is that man made climate change won't prevent the next ice age because the Earth will reabsorb the CO2 before then. He may be right, and I think that's a very sound and fair point to make. He's not actually saying an ice age will begin in about 2,000 years, though he does seem to imply the possibility.
I think, 2,000 years is likely too soon. The triggering of ice ages requires cold northern hemisphere summers (He says that in his article too). Basically, the red line on this chart should be low, and we're in a leveling off and small warming period for the next 50,000 years. The cycles are adding up in such a way that predicting the next ice age is hard.
On his predictions:
Predicting that mankind will run out of fossil fuels in 300 years is reasonable. It's something that can't be predicted with accuracy and I personally think it's wrong. I think we'll move away from coal as climate change gets worse and with abundant deep shale that can be fracked (natural gas is good for cooking and planes need fossil fuels), so I think fossil fuels will remain in use and to some extent, untapped considerably past 300 years, but my opinion aside, his prediction is still reasonable and possible.
Predicting that it will take the Earth about 2,000 years to reabsorb the added CO2 is also reasonable. I don't think anyone knows how long, or if a new equilibrium will be reached as opposed to a return to how it was, and making this kind of prediction is replete with uncertainties, but 2,000 years is fair. We should keep in mind that a science based hypothesis can be reasonable and still end up being wrong. In science as "good" hyopthesis is one that can be tested. A "good" hypothesis can still be wrong and for estimates this far out, there's room for both his estimate being reasonable and his estimate being very far off the mark. People sometimes yell at scientific predictions for being wrong, but predicting the future is hard, and people should stop yelling about that because it's not helpful. Scientists know that not all hypothesis based predictions are accurate, what matters is that a prediction is science based, has correlated cause and effect and can be studied and tested. (I hope you forgive my little rant, but I wanted to say what a good hypothesis actually means).
Some studies on oceanic absorption of CO2 and other estimates of how long it would take for the Earth to recover might improve this answer, or be the subject for another question, but I'm not feeling the urge right now. That's a tricky thing to estimate and 2,000 years seems fair enough to me.
It's funny that, today, it's in our best interest to try to stop using fossil fuels, but if/when the next ice age is coming, people might be encouraged to use clean fossil fuels to release more CO2 and prevent the coming ice age. "go for a drive, it's good for the environment" . . . anyone who's still around in 100,000 years, feel free to put that on your bumper-sticker. The author also indirectly makes the point that we might want to save some of our fossil fuels as a means of ice-age prevention. It's a sound point.
I think the article should have been longer and more fleshed out, but it's a good basic article.
I also want to add that while the article is short, he has a 267 page e-book which I imagine, goes into more detail. https://books.google.com/books/about/Energy_Demand_and_Climate_Change.html?id=EdAgjDtBsNsC