Short answer: No
Criteria has Querydsl-Like syntax which allows the generation of type-safe queries. Reading from Document may introduce MongoDB incompatible commands or syntax for the current MongoDB version.
Long answer: Yes
We can implement walkaround with Java reflection to add Document key:value into Criteria private fields.
Warning: You assume your Document has compatible operators with correct syntax.
Add this helper method into your class (souce):
public static Criteria from(Document document) {
    Criteria c = new Criteria();
    try {
        Field _criteria = c.getClass().getDeclaredField("criteria");
        _criteria.setAccessible(true);
        @SuppressWarnings("unchecked")
        LinkedHashMap<String, Object> criteria = (LinkedHashMap<String, Object>) _criteria.get(c);
        for (Entry<String, Object> set : document.entrySet()) {
            criteria.put(set.getKey(), set.getValue());
        }
        Field _criteriaChain = c.getClass().getDeclaredField("criteriaChain");
        _criteriaChain.setAccessible(true);
        @SuppressWarnings("unchecked")
        List<Criteria> criteriaChain = (List<Criteria>) _criteriaChain.get(c);
        criteriaChain.add(c);
    } catch (Exception e) {
        // Ignore
    }
    return c;
}
Now create new instance for Criteria from Document:
Criteria newCriteria = from(criteria.getCriteriaObject());
//You have full-compatible Criteria instance
newCriteria.and("foobar").elemMatch(Criteria.where("field").is("value"));