Would India have abolished caste slavery if the British had not invaded the country?Most likely, yes.
Consider the history of USA. They had slavery and oppression of women. And then they declared that all humans have some unalienable rights, and that one should not be taxed without representation. It was extremely difficult to justify slavery and oppression of women when such doctrines were in place. And today, there are demands for rights from the LGBT community, and several of their states have granted equal civil rights. Because no one can justify any of that discrimination under the rights they declared.
The case with India would have been the same. Though far from Europe, where movements to overthrow monarchy first gained momentum, India still had a lot of relations with the continent. And even if western European nations had not invaded, people would still have learnt of what was going on. Eventually, someone would have come up with the idea of democracy. Kings would have to be overthrown- but kings said they had a "divine right to rule". Monarchy could be abolished only if people moved away from such religious proclamations- and it is highly likely that people would say that everyone is born equal and must be treated equal before law.
The only way to avoid these ideas from catching on would have been total isolation- like what's going on with North Korea now. But something like that would not have been possible for India, which was NEVER under a single ruler.
Even after the british invaded and indian independence, there is only 5% inter-caste marriages in India. It is not abolished in real life. Why would 'India' with its 650 small states would have abolished it?
Caste is not slavery.