Has anyone attempted to quantify the relative ambiguity of languages and to rank them?
Your timing is excellent! The most comprehensive study I've seen on this topic was published less than a week ago (at the time of writing).
The first question is, what is ambiguity? Pretty much every sentence has some sort of ambiguity in it: if I say "I like cats", does "like" mean similar to, or feel affection toward (a lexical ambiguity)? The syntax here disambiguates, makes it clear that the answer is feel affection toward.
So why do English-speakers consider this ambiguity acceptable? Why isn't it a problem to have one word that means both similar to and feel affection toward? Because the two words are generally used in such different syntactic contexts that it isn't an issue. There are only so many strings of phonemes to use, and reusing some makes the average word length shorter, so it takes fewer phonemes to convey a meaning.
For a cross-linguistic example, Swahili has no separate words for "he", "she", "him", or "her": yeye is used regardless of case and gender. Ancient Greek has thirty (it just uses the demonstratives), allowing it to distinguish "the first man I mentioned" from "the second man I mentioned" from "a man newly introduced to the dialogue", along with five different grammatical cases (subject, direct object, indirect object, possessor, addressee).
So it seems that Swahili is significantly more ambiguous, right? It would need more words than Ancient Greek to convey the same amount of information: one Swahili word could correspond to any of thirty different Ancient Greek words, and only context can make it clear which is intended. Textbook ambiguity right there. That seems like something that can be objectively measured and ranked, using information theory.
Well, Coupé, Oh, Dediu, and Pellegrino did exactly that. (Less formal summary of their work here.) They calculated how many bits of information are contained in each word (and in each syllable) in seventeen different languages. Vietnamese, for example, conveys about eight bits per syllable, significantly more than English.
But when multiplied by the rate of speech—how many words/syllables are spoken per second—all the results were almost identical. Every single language they studied conveyed about 39.15 bits per second. Languages with a lower information density per word are spoken faster to compensate, getting in more words per second, while languages with a higher information density per word are spoken slower (since losing any individual word in transmission would be a bigger problem).
This implies that the answer to your question…
Are some human languages significantly less ambiguous than others?
…is no. Some languages may have more lexical ambiguities or more syntactic ambiguities, conveying less information per word. But they compensate by just using more words, putting in more context until the ambiguity is gone, in the same amount of time.