Good thing I never suggested that. Gentrification has a racial dimension because race correlates with economics, but it simply is the rich displacing the poor in a particular region; if you take housing units by eminent domain and establish a process for renting them out as public housing that doesn't distribute them to the highest bidder, you prevent gentrification. You neither have to acquire nor distribute based on race.
You are incorrect. Laws may both explicitly (or otherwise intentionally) target race and may disproportionately impact race without explicit targeting.
Laws doing the former are subject to “strict scrutiny”: the discrimination must be the least invasive means of achieving a compelling government interest. The latter isn't prohibited at all, though it can be evidence of discriminatory intent. (You may be thinking of employment law, where disparate impact is generally prohibited discrimination, unless closely tailored to a specific legitimate non-discriminatory business need.)
See https://en.m.wikipedia.org/wiki/Equal_Protection_Clause for a discussion, especially the section under “tiered scrutiny” and “disparate impact”.
and the process for renting them as public housing will be fairer how? You're just switching out one filtering system (price) for another based on arbitrary rules proposed by petty bureaucrats and politicians. At the end of the day you're still discriminating. The only difference that in your system, you're hoping that you or someone with your sensibilities has the power to do the discriminating.