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How can I invest in an index fund but screen out (remove) certain categories of socially irresponsible investments?
It sounds like you need an index fund that follows so called Sustainability index. A sustainability index does not simply select "socially responsible" industries. It attempts to replicate the target market, in terms of countries, industries, and company sizes, but it also aims to select most "sustainable" companies from each category. This document explains how Dow Jones Sustainability World index is constructed (emphasis mine): An example of a fund following such index is iShares Dow Jones Global Sustainability Screened UCITS ETF, which also excludes "sin stocks".