In South America and the Philippines, which were areas of Spanish/Portuguese colonization, the population is typically partially or fully native yet semi-integrated into Iberian culture (they are usually Catholic and usually speak Spanish/Portuguese).
In areas of British colonization such as New Zealand/Australia/US/Canada, the population is mostly of European descent.
Why is this? Is it entirely a result of higher population density in South America and the Philippines?
Was there less stigma on having children with the native people in Spanish colonies?
Was there some population phenomenon in Britain which meant that there were more women/families willing to move to the colonies? Were Iberian colonists mostly men, or were there large numbers of women/families who moves to the colonies as well? And if so, why did Iberian men marry native women more frequently than British colonists? Or is that assumption wrong, and British colonists frequently took native wives?
Were the British more brutal than the Iberians when it came to displacing native people?
While someone else with deeper knowledge on this issue comes, I can answer one of your questions:
Was there less stigma on having children with the native people in Spanish colonies?
We have to distinguish between actual marriages between Spaniards and natives and other forms of producing mixed children (largely rape). In the case of actual marriages, they usually involved the natives who had converted, either by choice or force, to Catholicism. Those who had adopted Catholicism and Spanish culture were called gente de razón ("people of reason"), and those that had not were simply considered uncultured or simply savages, and of course they were stigmatised.
It is very important to clarify that these interracial marriages and especially the children that came out of it were certainly not stigma-free. The Spanish Empire developed a very complex racial classification system called Casta, which gave different categories to children born to parents of different races. You had different rights and could face more or less racism (which was very prevalent in the Spanish empire) depending on which category you were placed.
The Casta system is very complex and it includes all possible racial combinations, and you have to think of it as a racial pyramid, with the most privileged classes at the top. At the top obviously were the Spaniards themselves, divided into two categories: the peninsulars (born in Spain) and the criollos (born in the Americas to peninsular parents). Despite being racially European, criollos were excluded from high-level positions in the government, just for the fact of them being born in the Americas. So you can imagine that the lowest you get in the Casta system, the worse off you are.
Why is this? Is it entirely a result of higher population density in South America and the Philippines?
Not to give the impression that this is a full, comprehensive answer to your question, but this is a very true statement. Populations of indigenous Americans were far higher in Central and South America than in what is now the United States and Canada. Pre-Columbian population estimates in Canada top out around 2 million... less than half of the lower end of population estimates for the Aztec empire alone, and that's just one of many cultural polities that existed in Central America at the time.
The places that you are thinking of as involving integrated populations likely had higher indigenous population density to begin with. Meanwhile, there are parts of Latin America where a particularly high percentage of the population is descended solely from Europeans, e.g. Argentina, which encompasses an area that was comparatively sparsely populated when Europeans arrived.