You need to provide more information. Whether the child is automatically a U.S. citizen at birth depends on the details of the situation:
- If the child was born in wedlock, and the U.S. citizen parent was physically present in the U.S. for 5 years before the child's birth, including 2 after turning 14, the child is automatically a U.S. citizen at birth. Nobody has the ability to "refuse" this.
- If the child was born out of wedlock, and the U.S. citizen parent was the mother, if the mother was physically present in the U.S. for 1 continuous year any time in her life before the child's birth, the child is automatically a U.S. citizen at birth. Nobody has the ability to "refuse" this either.
- If the child was born out of wedlock, and the U.S. citizen parent was the father, then in addition to the physical presence requirement above (physically present in the U.S. for 5 years before the child's birth, including 2 after turning 14), a few other conditions need to be met for the child to be considered a U.S. citizen from birth:
- A blood relationship between the child and the father is established by clear and convincing evidence
- The child’s father (unless deceased) has agreed in writing to provide financial support for the child until the child reaches 18 years of age
- One of the following criteria is met before the child reaches 18 years of age:
- The child is legitimated under the law of his or her residence or domicile;
- The father acknowledges in writing and under oath the paternity of the child; or
- The paternity of the child is established by adjudication of a competent court.
In the first two cases, the child is automatically a U.S. citizen from birth, and the desire of either parent has no effect on it. Any parent (not necessarily a U.S. citizen parent, and doesn't even need custody) or legal guardian can apply for the Consular Report of Birth Abroad. See here, under 7 FAM 1443.1:
d. The Form DS-2029 is to be executed only by a child’s parent(s) or
legal guardian(s). Either parent, including an alien parent, may
execute the application. [...]
f. Custody dispute: There is no two-parent signature requirement for a
Form FS-240 with respect to permission to issue. [...] In the case of
a child involved in a custody dispute, either parent may apply for the
Form FS-240 regardless of which parent has been awarded custody
(Of course, it may be harder to get evidence of the U.S. citizen parent's physical presence in the U.S. without their cooperation. But you may be able to get some evidence from public records, etc.)
If the child is over 18, you cannot apply for a Consular Report of Birth Abroad, but the child can just apply for a U.S. passport and/or a Certificate of Citizenship.
However, in the third case above (born out of wedlock to U.S. citizen father), the father's refusal to participate could mean the child is not a U.S. citizen. In particular, you need the father to agree in writing to support the child until 18. See here, under 7 FAM 1133.4-2(b)(3)
(3) Father's Statement of Support
(a) A statement of financial support
is required except when the father is deceased. A father who refuses
to sign a statement of support prevents his child from acquiring U.S.
citizenship. A child who cannot present a written support agreement by
the father cannot be documented as a U.S. citizen unless it is proven
that the father is dead. This is true even if the father cannot be
located; unless dead, the father must be located and comply with the
requirements of section 309(a), as amended, before the child's 18th
(b) Since section 309(a) specifies that the father must agree
in writing to support the child, a local law obliging fathers to
support children born out of wedlock is not sufficient to meet the
requirement of that section.