Gilbert Ryle noted ‘the rule-governed concatenation of questions, answerable by 'yes' or 'no,' which are intended to drive the answerer into self-contradiction’. He held that this ‘is what should be meant by 'the Socratic Method.' (Gilbert Ryle, Plato's Progress (Cambridge, 1966: 119.)
There are good reasons to think that Socrates uses a variety of methods of argument, so that there is no such thing strictly as ‘the Socratic Method’. I will return to this point.
I assume the quoted passage in the text box is from Euthyphro, 7d – 8a. However, that passage is not quite as you render it. The claim under review – a definition of piety - is that what is agreeable to the gods is pious and right, and what is disagreeable to the gods is impious and wrong (7a). The argument goes that disagreements among human beings are most likely over ethical questions rather than questions of magnitude, number, &., which can be easily settled. It is a fair inference, therefore, that if the gods disagree it is over the same questions. It is accepted that the gods do disagree (7e – 8a). And since they disagree, what is regarded as pious and right by one god is regarded as impious and wrong by another. In that case the same action is both pious and right as agreeable to one god and impious and wrong as disagreeable to another. The definition of piety has in the process of argument yielded a contradiction.
You are perfectly right that the effectiveness and cogency of this method of argument are situationally and personally specific. Contradiction will not follow, or a different contradiction will follow, depending on the answers the interlocutor gives. Socrates has no control over this. Euthyphro might not have accepted that the gods disagree over anything. In that case, the argument would have had a different outcome.
The general assumption behind Socrates’ use of this method of argument - ‘the rule-governed concatenation of questions, answerable by 'yes' or 'no’ - is that people do not know what they think they know. (This assumption is evidenced in Apology, 21e – 22e.) If Euthyphro had really known the nature of piety, he would not have run into the contradiction that hits him at the end of the argument.
The argument you quote is an example of eristic concluding in elenchus (refutation). This by no means covers and includes the range of argumentative method deployed by Socrates, who sometimes (a) uses dialectic to chart a progression of hypotheses ‘until you come to something acceptable’ (Phaedo, 101d) and (b) uses elenchus to positive effect as in the case of the slave body in the Meno, 80e – 86c, whom Socrates questions maioutically (like a midwife) to draw from him the solution to a mathematical problem, with the implication that the boy must have acquired the requisite knowledge in a pre-natal state; anamnesis (recollection) is evidently at work since Socrates’ questions have not conveyed any knowledge. (This is not to accept Socrates’ account of things; I merely indicate a method of argument.)
G. Ryle, Plato's Progress, Cambridge: CUP, 1966.
Plato: Complete Works, ed. J.M. Cooper & D.S. Hutchinson, Indianopolis: Hackett, 1997.