No, gay days does not push things in the other direction. Gay Days is essentially Orlando's Pride. June is Pride month in most places. You mention African American days...we also have Black History Month in February. Native American Heritage Month is in November. There are certainly other days/weeks/months where other groups celebrate their history and community, and take time to have perspective on where they have come from, the progress they have achieved, and to look forward to the future. If other groups want to have their own special day, no one is stopping them.
Gay Days/Pride (and other history months mentioned above) are for a traditionally marginalized community. LGBTQ people still do not have equal protections in the US in many aspects of law, and certainly in countries around the world where homosexuality is still considered a crime. In some places, still punishable by death. In the US, an employer can still fire someone for being gay in more than half of our states. Housing discrimination laws are similarly lacking. Hate crimes have been on the rise in several many major cities as of late. Trans people, especially trans women of color, face an insane level of violence and abuse. LGBTQ youth are still regularly shunned by family and kicked out of homes, and as much as 40% of the homeless population is estimated to be LGBTQ. Just the other day, a city worker started hurling anti-gay insults and the "other" f word at me in NYC...in a gayborhood. Discrimination still exists. People who do not belong to a marginalized group may not have experiences of discrimination like these, solely for being who they are. That's why majority groups tend not to have "special days," since they do not constantly have to fight for their right to exist.
My being gay is not the only facet of my life, but it has drastically altered my experiences: the way people interact with me, and thus the way I interact with people, it has formed a certain lens to life as I'm sure it has for many gay people because of their lived experiences. So yes gay days/pride is important, and no it doesn't push acceptance back. It is a time when I can remember those who came and fought and died before me. Where I can remind myself to cherish my community (past and present) and the parts of me that make me unique and have shaped my life experiences, instead of letting others highlight it as negative or wrong. I'm glad there is some type of Pride celebration at Disney where queer people can experience this for themselves and come together as a community in a joyous way.