Moral statements and actions are made everyday from scholars to everyday people. Morals can differ from person to person. What is moral to me may not be moral to you! I believe people make the best choices based on what is morally plausible! When people are not being morally just that is when that person falls into the wrong hands!
Moral realism means, according to Richard Boyd, that moral statements are either true or false and the approximate truth of those statements are independent of our moral opinions and theories. So it would be plausible and in fact, many philosophers today believe in moral realism which only gives it further credulity.
Moral realism is dependent on a cynical view of the world that retains the fixtures of morality. Being that morality is not a natural thing, because it changes from society to society, it is hard to make moral realism a big part of ones belief. But I believe that it is possible, because it has happened in society before.
Yes, I believe that moral realism is probable. The reason I believe such is because of the fact that we as society have broken and expanded the bounds of what was once considered "moral" and what is now moral. Though an individual's definition of moral is not the same as everyone else's, society dictates a standard of morality that is constantly being upheld by current standards of behavior, such as the fact that "to kill is immoral". During the time of Neanderthals, killing might have been considered necessary for survival, but certainly not moral. Thousands of years later, we hold ourselves to that very standard, which is proof that moral realism is probable.
Many try to use God as a justification for moral realism, but the theist is burden by the Euthyphro dilemma. To escape, they say Good is in God's Nature. This epistemically problematic and they would need to detail the morally good making property God has (which runs into issues with Divine Simplicty) or Good being in God's nature is arbitrary. Moral Naturalism runs into the naturalistic fallacy. Moral non-natural needs queer facts. Virtue ethicists have similar issues the theists as to how the Good is Good.
I think that moral realism is a nice concept in theory, but something that falls very far short in execution in the real world. It is quite similar to communism in this regard in that it sounds nice but can never really work. People and cultures are far too different.