Do false accusations and lying to the court automatically deem a parent unfit?

This is a general answer based on my experience in Santa Clara and Alameda Counties. I would be surprised if it's different in Solano County, but you should consult an attorney in your county to be sure.

The judge's primary focus in making the custody order will be the child's best interest. Generally, it is a guiding principle that a child needs time with BOTH parents, unless it is unsafe for the child. In this context, "unsafe" usually refers to abuse or neglect. Would be surprised to see a judge deny a parent visitation altogether for something like lying.

One thing a judge will consider in deciding who gets PRIMARY physical custody is which parent is MORE likely to foster continuing meaningful contact with the other parent. So, a parent who wants primary custody would be well advised to make it clear that he or she will ensure the child has appropriate continuing contact with the other parent--and then after getting primary custody, make good on his or her word (because a parent with primary custody can lose it by denying the other parent access unreasonably.) Where it has been exposed that one parent lied and abused the system to unreasonably deny the other parent access to the child, the judge is likely to consider that a factor in the falsely-accused parent's favor. In my own opinion, the falsely-accused parent would be well advised to refrain from saying things that sound like an attempt to deny the other parent access to the child. Again, among other factors in deciding who will have primary custody, the judge is looking for the parent who will be MORE reasonable and flexible in facilitating appropriate access for the other parent.

Regarding being "unfit" (which is a word I personally have not seen used in a custody order), I'm not aware of any "automatic" deeming of a parent to be unfit. If it isn't safe for the child to be alone with one parent because of abuse or neglect, I believe most judges would want to order parenting classes and supervised visitation before denying contact altogether.

Note: I practice is California, and this information pertains to California. This is general information, not legal advice tailored to a specific situation, and no attorney-client relationship is formed.