Python 3, 46 bytes

k=P=1
while k<1e6:P%k and print(k);P*=k*k;k+=1

By the time the loop reaches testing k, it has iteratively computed the squared-factorial P=(k-1)!^2. If k is prime, then it doesn't appear in the product 1 * 2 * ... * (k-1), so it's not a factor of P. But, if it's composite, all its prime factors are smaller and so in the product. The squaring is only actually needed to stop k=4 from falsely being called prime.

More strongly, it follows from Wilson's Theorem that when k is prime, P%k equals 1. Though we only need that it's nonzero here, it's useful in general that P%k is an indicator variable for whether k is prime.

Python 3, 45 bytes

k=P=1
while k<1e6:P%k>0==print(k);P*=k*k;k+=1

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By the time the loop reaches testing k, it has iteratively computed the squared-factorial P=(k-1)!^2. If k is prime, then it doesn't appear in the product 1 * 2 * ... * (k-1), so it's not a factor of P. But, if it's composite, all its prime factors are smaller and so in the product. The squaring is only actually needed to stop k=4 from falsely being called prime.

More strongly, it follows from Wilson's Theorem that when k is prime, P%k equals 1. Though we only need that it's nonzero here, it's useful in general that P%k is an indicator variable for whether k is prime.

Thanks to @Sisyphus for 1 byte with P%k>0==print(k) using chained operator short-circuiting in place of P%k and print(k).

Python 3, 46 bytes

k=P=1
while k<1e6:P%k and print(k);P*=k*k;k+=1

By the time the loop reaches testing k, it has iteratively computed the squared-factorial P=(k-1)!^2. If k is prime, then it doesn't appear in the product 1 * 2 * ... * (k-1), so it's not a factor of P. But, if it's composite, all its prime factors are smaller and so in the product. The squaring is only actually needed to stop k=4 from falsely being called prime.

More strongly, it follows from Wilson's Theorem that when k is prime, P%k equals 1. Though we only need that it's nonzero here, it's useful in general that P%k is an indicator variable for whether k is prime.

Python 3, 46 bytes

k=P=1
while k<1e6:P%k and print(k);P*=k*k;k+=1

By the time the loop reaches testing k, it has iteratively computed the squared-factorial P=(k-1)!^2. If k is prime, then it doesn't appear in the product 1 * 2 * ... * (k-1), so it's not a factor of P. But, if it's composite, all its prime factors are smaller and so in the product. The squaring is only actually needed to stop k=4 from falsely being called prime.

More strongly, it follows from Wilson's Theorem that when k is prime, P%k equals 1. Though we only need that it's nonzero here, it's useful in general that P%k is an indicator variable for whether k is prime.

Python 3, 46 bytes

k=P=1
while k<1e6:P%k and print(k);P*=k*k;k+=1

By the time the loop reaches testing k, we've computed the product P=(k-1)!^2. k is prime only if it isn't a factor of this number, since otherwise its prime factors will already have appeared in the product. The squaring is only actually needed to stop k=4 from falsely being called prime.

More strongly, it follow from Wilson's Theorem that when k is prime, k!^2 equals 1. While we only need that it's nonzero here, it's useful in general that P%k is an indicator variable for whether k is prime.

Python 3, 46 bytes

k=P=1
while k<1e6:P%k and print(k);P*=k*k;k+=1

By the time the loop reaches testing k, it has iteratively computed the squared-factorial P=(k-1)!^2. If k is prime, then it doesn't appear in the product 1 * 2 * ... * (k-1), so it's not a factor of P. But, if it's composite, all its prime factors are smaller and so in the product. The squaring is only actually needed to stop k=4 from falsely being called prime.

More strongly, it follows from Wilson's Theorem that when k is prime, P%k equals 1. Though we only need that it's nonzero here, it's useful in general that P%k is an indicator variable for whether k is prime.