Sperm are unquestionably alive, according to all or most sensible definitions of life (They move around, they have goals, they eat things, they die). They are not, however, a human or animal life.
Vineet Menon has a shaky grasp of species definitions, and I'd like to correct some misunderstandings. Chromosome number is not that useful in distinguishing different species. There are a large number of genetic disorders that alter your chromosomal count without making you not a human being. (Klinefelter, Turner, Edward, Down, etc.) It's also worth noting that all mosses are haploid except during the sporophyte stage of their life cycle, when they're diploid. I doubt Vineet Menon intends to categorically declare mosses 'not alive'.
Your blood is alive, and your feces is also thriving with bacteria. Some yogurt is alive, most cheese is alive. All fermented beverages are alive, or were once. Bread dough is alive. The argument you linked to is debating whether or not they are alive enough to qualify as a macroscopic living thing for ethical purposes, which is an entirely different question. A chicken definitely qualifies, single-celled algae like spirulina, mushrooms, and yeast do not. Eggs, fertilized or not, seem to be somewhere in the middle. It is a continuous spectrum with people and cats and chickens and things at one end, and viruses and tuberculosis and dirt at the other. Eggs and sperm and mushrooms I would put somewhere in the middle.
So whether the answer to the question 'Are gametes alive?' is unquestionably yes;
The answer to the question of whether or not that makes them 'a life' is up to you.