Later Wednesday, Herring voluntarily stepped aside as co-chair of the Democratic Attorney Generals Association for an interim period.
Northam, also Democrat, still faces widespread calls for his resignation after a conservative news site last week published his yearbook page photo.
And Northam's would-be successor, Lt. Gov. Justin Fairfax, another Democrat, was accused earlier this week of sexually assaulting a woman at the Democratic National Convention in 2004. He denies her accusation.
If Northam, Fairfax and Herring all resign, the state's speaker of the House of Delegates, Kirk Cox, would become governor. Cox is a Republican.
The Republican majority in the House of Delegates was decided by a name being drawn from a bowl to break a tie in a race for one seat in that legislative body.
A Democrat has been in Virginia's governor's mansion since 2014.
Sen. Mark Warner, D-Va., said he was "shocked and disappointed" by Herring's disclosure.
Warner said the week has been an "awful" one for Virginia.
He refused to comment further when asked by NBC News if Herring should resign.
On Thursday, after this article was first published, Warner's spokeswoman emailed CNBC to say that Warner had not called for Northam's resignation because of the possibility that he may have worn blackface, as NBC News's question to him had suggested he did.
Instead, the spokeswoman pointed to the statement that Warner issued last Saturday with Sen. Tim Kaine, D-Va., and Rep. Bobby Scott, D-Va., in which they said: "After we watched his press conference today, we called Governor Northam to tell him that we no longer believe he can effectively serve as Governor of Virginia and that he must resign."
"Governor Northam has served the people of the Commonwealth faithfully for many years, but the events of the past 24 hours have inflicted immense pain and irrevocably broken the trust Virginians must have in their leaders. He should step down and allow the Commonwealth to begin healing."
In his full statement Wednesday, Herring said:
"The very bright light that is shining on Virginia right now is sparking a painful but, I think we all hope, important conversation. The stakes are high, and our spirits are low.
"I am sure we all have done things at one time or another in our lives that show poor judgment, and worse yet, have caused some level of pain to others. I have a glaring example from my past that I have thought about with deep regret in the many years since, and certainly each time I took a step forward in public service, realizing that my goals and this memory could someday collide and cause pain for people I care about, those who stood with me in the many years since, or those who I hoped to serve while in office.
"In 1980, when I was a 19-year-old undergraduate in college, some friends suggested we attend a party dressed like rappers we listened to at the time, like Kurtis Blow, and perform a song. It sounds ridiculous even now writing it. But because of our ignorance and glib attitudes — and because we did not have an appreciation for the experiences and perspectives of others — we dressed up and put on wigs and brown makeup.
"This was a onetime occurrence and I accept full responsibility for my conduct.
That conduct clearly shows that, as a young man, I had a callous and inexcusable lack of awareness and insensitivity to the pain my behavior could inflict on others. It was really a minimization of both people of color, and a minimization of a horrific history I knew well even then.
"Although the shame of that moment has haunted me for decades, and though my disclosure of it now pains me immensely, what I am feeling in no way compares to the betrayal, the shock, and the deep pain that Virginians of color may be feeling. Where they have deserved to feel heard, respected, understood, and honestly represented, I fear my actions have contributed to them being forced to revisit and feel a historical pain that has never been allowed to become history.
"This conduct is in no way reflective of the man I have become in the nearly 40 years since.
"As a senator and as attorney general, I have felt an obligation to not just acknowledge but work affirmatively to address the racial inequities and systemic racism that we know exist in our criminal justice system, in our election processes, and in other institutions of power. I have long supported efforts to empower communities of color by fighting for access to healthcare, making it easier and simpler to vote, and twice defended the historic re-enfranchisement of former felons before the Supreme Court of Virginia. I have launched efforts to make our criminal justice system more just, fair, and equal by addressing implicit bias in law enforcement, establishing Virginia's first-ever program to improve re-entry programs in local jails, and pushing efforts to reform the use of cash bail. And I have tried to combat the rise in hate crimes and white supremacist violence that is plaguing our Commonwealth and our country.
"That I have contributed to the pain Virginians have felt this week is the greatest shame I have ever felt. Forgiveness in instances like these is a complicated process, one that necessarily cannot and should not be decided by anyone but those directly affected by the transgressor, should forgiveness be possible or appropriate at all. In the days ahead, honest conversations and discussions will make it clear whether I can or should continue to serve as attorney general, but no matter where we go from here, I will say that from the bottom of my heart, I am deeply, deeply sorry for the pain that I cause with this revelation.