"Greater acceptance" means that more people are willing to embrace or be open to something/someone, often despite differences or previous resistance. Full definition
We are fortunate enough to live in a much more tolerant society, with greater acceptance of different ways of life. (totalpolitics.com)
In my work with couples I frequently hear one or both partners state the need for greater acceptance in their relationship or marriage. (strengthenyourrelationship.com)
To gain greater acceptance of standard forms, be flexible with alternate language to suit each lawyer's style. (joelarose.com)