Is it possible to grow a fertilized human egg entirely outside the woman's womb?
I believe what you're describing is referred to as ectogenesis which is essentially developing a fertilized egg inside an artificial womb until it’s able to survive outside the controlled environment. So is this possible outside of science fiction? To answer this we have to compare the stages of human reproduction and how close today's science and technology can come replicating/sustaining them. 

We'll begin by dividing these stages into two groups: fertilization and development of embryo to fetus.

When it comes to answering the question of whether fertilization can be replicated in vitro or not, the answer is yes it can be done! In fact in vitro artificial fertilization (IVF) has been possible for several decades. In most cases when IVF is performed the egg is then placed inside the mother's uterus where it goes onto to develop into an embryo and then fetus. In regards to "ectogenesis" the IVF egg would be placed into an artificial womb.

In order to form a zygote (fertilized egg) to develop properly into a fetus it has to be an environment to meet it needs. In a female uterus all these needs are met but replicating them might be difficult. An artificial womb would have to be able to provide nutrients, oxygen, and channels for the development process of a fetus as well as system to expel (birth) the fetus once its development is complete. So can today's technology and science do all this? In theory yes, if resources and time where dedicated (and red tape cut) it would be possible to develop a fertilized egg in an artificial environment within the foreseeable future. There have been multiple experiment where artificial wombs were implanted with fertilized eggs and began to grow but were stopped due to legality.

I encourage you to read the following Wikipedia and motherboard articles regarding artificial wombs and ectogenesis. 

https://en.wikipedia.org/wiki/Artificial_uterus

http://motherboard.vice.com/read/artificial-wombs-are-coming-and-the-controversys-already-here

*Note there is a lot of biology I did not mention regarding zygote to fetus development which is the biggest question/obstacle that ectogenesis might face