During the colonial era Indië or Oostindië was used in Dutch for a large part of Asia. E.g. the Dutch East India Company VOC (Vereenigde Oostindische Compagnie) carried out all Dutch colonial activities in Asia.
Also you can come across Indië for Indonesia, as it is a short name for Nederlands-Indië, the name of Indonesia when it was a Dutch colony (until 1949). As the generation that grew up with this Indië is dying out, this won't be used much anymore, except for historical contexts.
Because of this, I wouldn't advise using Indië for India, as you'll likely to cause confusion as people will probably think you mean one of the 2 definitions I mentioned above.
On the upside…in Dutch there is not confusion between the Indians from America (indianen) and the ones living in India (Indiërs). But in a historical context again, Indiërs could also mean people from Nederlands-Indië.
Edit: what I said, combined with the vrt link by lobbens basically is the whole explanation. :)
It seems you are correct. I grew up being taught that term and accordingly used it wrongly ever since. There seems to be more to it though. We still use derivatives of 'Indië' in names like 'Indische Oceaan' or 'Indiër', but we indeed have to say 'India' and the 'Indiase regering' —http://www.vrt.be/taal/india-indi%C3%AB