Well, the short answer is no. Both animals and plants have evolved extensive mechanisms which prevent this from happening.
First, sperm have to find their way to an egg. In mammals, when the egg is released from the ovary it is surrounded by a loose group of follicle cells. These cells release a chemical signal that the sperm swim toward. If the signal is not correct, the sperm won't find the egg.
While the sperm are swimming up the female's reproductive tract, ions that her body secretes are absorbed into the sperm cells, causing internal changes that are referred to as capacitation. In humans, capacitation takes 5 to 6 hours and must be completed before the sperm can accomplish its next job, which is to penetrate the zona pellucida, or egg coat. The zona pellucida has proteins that are similar to the proteins used by the immune system, and these proteins make cross-species fertilization nearly impossible.
Once the sperm has successfully penetrated the zona pellucida, the acrosome reaction allows the sperm to fuse with the membrane of the egg so that the male's DNA can enter the egg cell. The acrosome reaction requires specific proteins that again prevent hybridization between species.
In order for cell division to occur and create an embryo, the paternal and maternal DNA must line up - another block to hybridization, as DNA varies from species to species enough for this to be unlikely to work.
An interesting side note is that fertility clinics sometimes use enzymes to strip the zona pellucida away from hamster eggs, and then use the stripped eggs to test human sperm for penetrating power. Although healthy sperm can penetrate the hamster eggs in this case, the DNA alignment fails and the fertilized eggs die without undergoing mitosis.