No, unlike many other Indo-European languages, current English has no default gender. Grammatically speaking, English does not have gender at all—the only gender that English marks at all is natural gender (sex).
When the natural gender of a living being is known, it is customary to refer to that being using pronouns that reflect that gender; for male beings, we use he/him/his, and for female beings, we use she/her/her(s). As you say, animals do have natural gender, and there is nothing to stop you from using the corresponding gendered pronoun. When referring to your own dog, for example, most people will use gendered pronouns. In many cases, however, you simply don’t know what sex an animal is at first blush.
With some animals, the distinction is fairly easy to make—a buck with its antlers looks quite different from a doe, for instance—and figuring out the right pronoun tends to be easy. With other animals, however, it’s not so easy. I for one would have absolutely no clue whether a cat or a turtle or a grizzly bear is male or female just by looking at it.
In those cases, you can either choose to use a gendered pronoun and then have a 50% chance of being right; or you can skirt the issue altogether and use it, which is sex-agnostic.
The same is true of human infants, incidentally. Babies all look the same (quoth the bachelor), and it can be quite impossible to tell whether a clothed baby is a boy or a girl. Until you find out, you can dodge the risk of mis-gendering it by calling it it (cf. this question).
Historically speaking, English did of course have genders, including using the masculine as the default gender; but while some still consider using he to refer to being(s) of unknown sex to be neutral, many people nowadays consider this to go against the notions of equal rights. Some have started using she as the default gender instead for obviously animate beings. When dealing with (non-infant) human beings, where it would be depersonalising and degrading, many now use singular they.
With animals, the notion of ‘depersonalising’ is rarely an issue, and it remains the more common way to get around the issue. That is, the ‘default gender’ for non-human entities is no gender, whether the entity in question has a natural gender or not.