The word integrity meaning wholeness seems to come from the word integer which, roughly speaking, refers to whole numbers. Why does integrity also mean honesty?
Etymology online has the following entry for integrity:
mid-15c., "wholeness, perfect condition," from O.Fr. integrité, from L. integritatem (nom. integritas) "soundness, wholeness," from integer "whole" (see integer). Sense of "uncorrupted virtue" is from 1540s.
That doesn't answer why integrity also means honesty, but you can see that circa 1540 the word began being used to express "uncorrupted virtue," which implies honesty, as a virtuous person does not lie.
I agree with @John Weldon that integrity and honesty aren't synonymous.
So defining integrity becomes important, and I'm inclined to go with the crowd on that--integrity in a human being is a state of having all the important virtues. Being "whole" or "complete" as a being of virtue. There would probably be some disagreement about which virtues are required for integrity, of course.
In this view, honesty can be seen as a necessary condition of integrity--you can't be "integrated" without it--but you can be honest and still lack integrity if you also lack other crucial virtues.
I think people sometimes think of integrity as synonymous with honesty because it's one of the most obvious virtues, and one that most people can agree is necessary for integrity.
This question ->1<- was closed, but I don't see it as a duplicate of this one at all. In the Mental Health professions, one speaks of "being Integrated" or "well Integrated", but the term "integrity" itself almost invariable has no special meaning. Rather, it has the common usage that associates it with ethical or moral values.
A person could be "well integrated" i.e., in no way suffering from Associative Identity Disorder (the current term, not M.P.D.), and still have ethics or morals that their society would judge to be poor. So, the answer to the closed question is No, 'Integrity' is not an antonym to 'Honesty', but 'Integrated' could certainly be considered one.
The answer to this question is that strictly speaking Integrity and Honesty are not identical, but in common usage what most other members of a society consider to be a whole, complete "well adjusted" person is usually one that they also consider honest.