Does a woman contain all the genes needed to make a man?
Human female cells contain most of the genetic information required to make a male, but they do not contain a critical component: The Y chromosome. This is a relatively small chromosome. Wikipedia claims we have identified around 200 genes on it to date, compared to estimates of 20,000 - 25,000 genes overall in the human genome. 

Importantly for your question, only other males contain and make full copies of all the genes on the Y chromosome. The Y chromosome does not cross-over* with the X chromosome during meiosis when producing sperm cells, which keeps the Y genes isolated, except for rare exceptions (e.g. gene translocation). So those genes are passed down the male line only. The SRY gene is just one of the genes encoded uniquely on that chromosome, so something else is definitely missing - the question could be re-phrased "how important to being a functional male are the unique genes other than SRY unique to the Y chromosome?"

The linked question (an experiment on mice that produced XX "males" by finding an alternative trigger to the SRY gene to induce male development) demonstrates that in terms what we might consider a physical male specimen, the Y chromosome probably contains a minority of critical unique features. The mice developed into apparently complete physical males. That means genes outside of the Y chromosome cover most of the body development plan for both males and females. 

However, such XX males will in most mammal species not be fully functioning males - the missing genes do make a difference. One outcome is infertility - biologically speaking, XX males do not possess a key male function, the ability to mate with a female to produce offspring. There are missing genes related to sperm production. 

Human XX males - as in developmentally male - do occur naturally. This can be caused by gene translocation - specifically the SRY gene moves to the X chromosome in a sperm from the father (potentially other Y genes will move in the same event). The symptoms that present with the syndrome vary, but give some indication of which critical components are missing.



Not all animals have the same mechanism. Birds have W and Z chromosomes (with males being ZZ and females ZW). Yet other animals do not use genetic switches, but environmental ones. In those cases, the female does contain all the necessary genetic information to create a male. In cases where there is parthenogenesis (production of offspring without fertilization), then this can be fully demonstrated.

* Technically there are some small regions, about 5% of the Y chromosome, which do cross-over with X, called the pseudoautosomal regions. At the time of writing the linked article mentions 29 genes found in these regions.