Your father can disinherit this child but he does need to specifically mention him in his testamentary document so that the adopted out son can not claim he is a pretermitted heir. (a child in which the law presumes that the testator accidentally forgot about). If there was no testamentary document, the son would only be an heir if he lived with your father for some period during his life since he was adopted out. You should make sue your father has the disinheritance clause drafted by an attorney.

First, as he was unrelated to your mother by birth or adoption he has no rights to her estate. No one has a right to inherit from either of their parents, any child may be disinherited. This applies both to children born within and outside of marriage. If your father does not execute a properly drafted will we look at California's intestacy statute to determine whether your half sibling has any rights. Although adoption ordinarily severs the parent child relationship for intestate succession purposes, there is an exception under 6451 of the California Probate Code which may apply to this potential competing heir. If your half-brother ever lived with your Dad, as your Dad's son, the adoption will be disregarded and he will have the same rights as he would have had if he was not adopted by another. All that said, your father can still disinherit him. I've attached code section 6451 into my answer so that you can read it yourself.

6451. (a) An adoption severs the relationship of parent and child between an adopted person and a natural parent of the adopted person unless both of the following requirements are satisfied:
(1) The natural parent and the adopted person lived together at any time as parent and child, or the natural parent was married to or cohabiting with the other natural parent at the time the person was conceived and died before the person’s birth.
(2) The adoption was by the spouse of either of the natural parents or after the death of either of the natural parents.
(b) Neither a natural parent nor a relative of a natural parent, except for a wholeblood brother or sister of the adopted person or the issue of that brother or sister, inherits from or through the adopted person on the basis of a parent and child relationship between the adopted person and the natural parent that satisfies the requirements of paragraphs (1) and (2) of subdivision (a), unless the adoption is by the spouse or surviving spouse of that parent.
(c) For the purpose of this section, a prior adoptive parent and child relationship is treated as a natural parent and child relationship.
(Added by Stats. 1993, Ch. 529, Sec. 5. Effective January 1, 1994.)

If he was adopted, he has no right to inheritance because he is no longer a 'legal' son.

I agree with Attorney Straus. Under California law, the son's adoption by his step-father legally severs any inheritance rights he would have had as to his biological father (your dad).
Assuming that your dad has capacity, your dad is free to include whomever he wants in his will.

If your father has acknowledged your half-brother as his son, the illegitimacy is irrelevant. And a step-parent adoption does not cut off inheritance rights the way a regular adoption does. So your father needs to decide how he wants to handle any inheritance to this son. He can leave him an inheritance if he wants. If he doesn't want to, he should specifically disinherit him.

This must be disconcerting for all of you. But this is entirely your father's decision. Just as he could leave an inheritance to a friend, a nephew, or a charity, he can leave an inheritance to this son you didn't know he had.

Your Dad may disinherit whomever he chooses.

No. Under California law, this cannot occur. That said, a properly drafted will won't be contested.

This is an estate planning question so I will put it there, but the other attorney is correct - he must be specifically named in the Will and then specifically have nothing left to him. If he is just omitted, he will have a claim to some of the estate. You need counsel from an experienced estate planning attorney.

All (note ALL) children must be named in the will as existing and then the Testator can leave whatever he chooses to any child. If he fails to name the son, regardless as to character, the son inherits. This position is based upon English Common Law and has been part of the law in the United States since there was a United States. You cannot omit naming heirs, you can only decline to give them anything.