Taking Australian history as the history recorded in an empirical fashion since 1788, and being "proud" or Australian history as looking back over that history with a sense of patriotism, then yeah I think it is quite possible to be proud of our history.
An essential feature of this debate should be a comparison with the individual histories of different nations across the globe. In comparison to the vast majority with a history of over 200 years, Australia is fairly clean. While the atrocities committed against Aboriginal people and the recent anti-Islamic violence stand out as blotches in Australian history, the Americans actually fought a full-scale war over slavery; man killed man defending his 'right' to enslave another human being. There is hardly a nation in the world that can claim to have possessing a spotless history, though in comparison, Australia comes out quite well.
Australian history has its very dark moments, the stolen generation and initial genocidal action towards Aboriginal people are pre-eminent, and should be taught in Australian history in schools for obvious reasons. But they do have to be viewed in a broader context - in 1967, a 98% majority voted to give citizenship rights to Aboriginal people, and in 1901 giving such rights to indigenous people was unheard of; it would have been nice to lead the way, but the fact that we didn't shouldn't prevent a positive reading of Australian history.
The simple fact is that Australian history since 1788 is overwhelmingly positive when viewed in a global context. And for that, we should be proud. One doesn't have to get overly nationalistic about it; our involvement in all wars has been of little significance - Kokoda was brave and undeniably brilliant, but Australia did not win WWII, the Allies (of which Australia was a minor member) did - and the Cronulla riots do little to present Australia to the world as a textbook multicultural society. But we are a nation with a fantastic individual culture and cultures, we're educated, prosperous and contributing more than our fair share to the global economy, and we're pretty good at swimming. For a nation that hit 20 million only recently, we're doing pretty well.
We have a strong history, with more highs than lows, and therefore I personally look back over the history of Australia since 1788 with some level of pride.