What does 'levitical' mean in this context?

I would assume it to be an allusion to the biblical Book of Leviticus which, in chapters 11–15, contains a long list of rules and rituals related to cleanliness and purification. Some of these, like the purification ritual described in Leviticus 14:1–32, are extremely elaborate, detailed and time-consuming:

¹ And the Lord spake unto Moses, saying, ² This shall be the law of the leper in the day of his cleansing: He shall be brought unto the priest: ³ And the priest shall go forth out of the camp; and the priest shall look, and, behold, if the plague of leprosy be healed in the leper; ⁴ Then shall the priest command to take for him that is to be cleansed two birds alive and clean, and cedar wood, and scarlet, and hyssop: ⁵ And the priest shall command that one of the birds be killed in an earthen vessel over running water: ⁶ As for the living bird, he shall take it, and the cedar wood, and the scarlet, and the hyssop, and shall dip them and the living bird in the blood of the bird that was killed over the running water: ⁷ And he shall sprinkle upon him that is to be cleansed from the leprosy seven times, and shall pronounce him clean, and shall let the living bird loose into the open field.

⁸ And he that is to be cleansed shall wash his clothes, and shave off all his hair, and wash himself in water, that he may be clean: and after that he shall come into the camp, and shall tarry abroad out of his tent seven days. ⁹ But it shall be on the seventh day, that he shall shave all his hair off his head and his beard and his eyebrows, even all his hair he shall shave off: and he shall wash his clothes, also he shall wash his flesh in water, and he shall be clean.

¹⁰ And on the eighth day he shall take two he lambs without blemish, and one ewe lamb of the first year without blemish, and three tenth deals of fine flour for a meat offering, mingled with oil, and one log of oil. ¹¹ And the priest that maketh him clean shall present the man that is to be made clean, and those things, before the Lord, at the door of the tabernacle of the congregation: ¹² And the priest shall take one he lamb, and offer him for a trespass offering, and the log of oil, and wave them for a wave offering before the Lord: ¹³ And he shall slay the lamb in the place where he shall kill the sin offering and the burnt offering, in the holy place: for as the sin offering is the priest's, so is the trespass offering: it is most holy: ¹⁴ And the priest shall take some of the blood of the trespass offering, and the priest shall put it upon the tip of the right ear of him that is to be cleansed, and upon the thumb of his right hand, and upon the great toe of his right foot: ¹⁵ And the priest shall take some of the log of oil, and pour it into the palm of his own left hand: ¹⁶ And the priest shall dip his right finger in the oil that is in his left hand, and shall sprinkle of the oil with his finger seven times before the Lord: ¹⁷ And of the rest of the oil that is in his hand shall the priest put upon the tip of the right ear of him that is to be cleansed, and upon the thumb of his right hand, and upon the great toe of his right foot, upon the blood of the trespass offering: ¹⁸ And the remnant of the oil that is in the priest's hand he shall pour upon the head of him that is to be cleansed: and the priest shall make an atonement for him before the Lord. ¹⁹ And the priest shall offer the sin offering, and make an atonement for him that is to be cleansed from his uncleanness; and afterward he shall kill the burnt offering: ²⁰ And the priest shall offer the burnt offering and the meat offering upon the altar: and the priest shall make an atonement for him, and he shall be clean.

²¹ And if he be poor, and cannot get so much; then he shall take one lamb for a trespass offering to be waved, to make an atonement for him, and one tenth deal of fine flour mingled with oil for a meat offering, and a log of oil; ²² And two turtledoves, or two young pigeons, such as he is able to get; and the one shall be a sin offering, and the other a burnt offering. ²³ And he shall bring them on the eighth day for his cleansing unto the priest, unto the door of the tabernacle of the congregation, before the Lord. ²⁴ And the priest shall take the lamb of the trespass offering, and the log of oil, and the priest shall wave them for a wave offering before the Lord: ²⁵ And he shall kill the lamb of the trespass offering, and the priest shall take some of the blood of the trespass offering, and put it upon the tip of the right ear of him that is to be cleansed, and upon the thumb of his right hand, and upon the great toe of his right foot: ²⁶ And the priest shall pour of the oil into the palm of his own left hand: ²⁷ And the priest shall sprinkle with his right finger some of the oil that is in his left hand seven times before the Lord: ²⁸ And the priest shall put of the oil that is in his hand upon the tip of the right ear of him that is to be cleansed, and upon the thumb of his right hand, and upon the great toe of his right foot, upon the place of the blood of the trespass offering: ²⁹ And the rest of the oil that is in the priest's hand he shall put upon the head of him that is to be cleansed, to make an atonement for him before the Lord. ³⁰ And he shall offer the one of the turtledoves, or of the young pigeons, such as he can get; ³¹ Even such as he is able to get, the one for a sin offering, and the other for a burnt offering, with the meat offering: and the priest shall make an atonement for him that is to be cleansed before the Lord. ³² This is the law of him in whom is the plague of leprosy, whose hand is not able to get that which pertaineth to his cleansing.

^{(Text from the public-domain King James Version, via biblegateway.com)}

Presumably the author of the paragraph that you quoted (who I'm pretty sure I recognize) was likening the meticulous cleaning procedures required when working with highly concentrated hydrogen peroxide to the ritual purification procedures described in this section of the Bible.