A multiverse provides the determinist with an explanation for randomness. This explanation hopes to avoid the suggestion that there exist events originating from the free choices of agents on statistical grounds.
First consider the anthropic principle. Here is Wikipedia's description:
The anthropic principle is a philosophical consideration that observations of the universe must be compatible with the conscious and sapient life that observes it. Proponents of the anthropic principle reason that it explains why this universe has the age and the fundamental physical constants necessary to accommodate conscious life. As a result, they believe it is unremarkable that this universe has fundamental constants that happen to fall within the narrow range thought to be compatible with life. [my emphasis]
Wikipedia links the section in bold with its "Fine-tuned universe" article.
The fine-tuned universe is the proposition that the conditions that allow life in the Universe can occur only when certain universal dimensionless physical constants lie within a very narrow range of values, so that if any of several fundamental constants were only slightly different, the Universe would be unlikely to be conducive to the establishment and development of matter, astronomical structures, elemental diversity, or life as it is understood.
A fine-tuned universe suggests there is either a tuner, which points to theism, or randomness. Randomness, by itself, is a threat to determinism. Also it is not an explanation. The multiverse attempts to provide that explanation.
Generally, whenever something appears 'random', and there is nothing to explain that randomness, that opens up a possibility that a free choice from somewhere occurred choosing one of the possibilities. Such a free choice coming from anywhere would falsify the hypothesis of determinism. To avoid that falsification, just add to the assumption of determinism another assumption that if scientific evidence forces one to accept randomness then all of the possibilities occur in some universe of a multiverse.
The OP asks whether determinism and a multiverse are contradictory.
To get a logical contradiction one needs two propositions, one being the negation of the other, that someone commits to. It may be easier to obtain a contradiction between determinism and randomness. Even there finding a logical contradiction may not be easy.
However, if one is opposed to determinism another approach taken by the Information Philosopher is to view determinism as a "dogma" that needs to be revised and then provide an alternative to it.
Wikipedia contributors. (2019, August 27). Fine-tuned universe. In Wikipedia, The Free Encyclopedia. Retrieved 13:30, September 5, 2019, from https://en.wikipedia.org/w/index.php?title=Fine-tuned_universe&oldid=912702879
Wikipedia contributors. (2019, August 29). Anthropic principle. In Wikipedia, The Free Encyclopedia. Retrieved 13:26, September 5, 2019, from https://en.wikipedia.org/w/index.php?title=Anthropic_principle&oldid=913044836