Yes, in some situations. A child can only have two legal parents. This means that the parent that is not your spouse must give up his or her parental rights to be guardian of his or her child. This can happen in a few different ways but generally this means that (1) the parent signs a piece of paper consenting to the adoption and terminating their own rights or (2) the child is abandoned. Abandonment is a legal term and to prove it, generally one must search for the father through legal channels. Abandonment can also be financial abandonment, by not supporting the child financially.
Many times a father or mother will give up their parental rights and sign a waiver because it allows that person to terminate child support as well. Other times, the parent will give up rights simply because they have never been involved in the child's life. In regards to giving up child support, that means that the adoptive parent has taken on the legal responsibility of financially supporting the child, and, as part of the proceedings, the adoptive parent must attest to his or her ability to financially support the child.
There is also a requirement that the adoptive parent be married to the child's parent (your spouse) and have lived with the child for six months preceding the actual adoption. This requirement has to do with showing the stability of the relationship. As we all know, divorce happens in a great number of marriages. Becoming an adoptive parent is not something to take on lightly. In the event of a divorce, you have parental rights equal to the natural mother or father of the child. This means that you also have custody rights in the event of a divorce.
One more requirement is a background check. It is entirely possible that a past crime, especially any child abuse or neglect, could bring a proceeding to a halt and bar you from becoming an adoptive parent.
In some jurisdictions, a home study is required and a Guardian Ad Litem will talk with both your spouse and you, the prospective adoptive parent. A Guardian Ad Litem is an attorney looking out for the best interest of the child. This person will have to make a recommendation for or against the adoption. Generally, in the home study, this attorney is looking for safety and cleanliness of the home the child is to be raised in. In all jurisdictions, the Guardian Ad Litem will conduct an interview of both of the parents as well as the child (although if the child is very very young the child may not be interviewed-usually within the discretion of the GAL, but somewhere between the ages of 0-4).
Bottom Line: Most conscientious people will make the grade and get the approval of the court and the Guardian Ad Litem. The place where some step-parent adoptions get unhitched is in the early stages, when the other parent (non-spouse) does not want to give up parental rights.