Iran is a fascinating study. The answer is "no", but it is a question which deserves more than a binary answer. Western powers for various reasons of self-interest, recent history and regional alliances seek to suppress Iran, so the impression of Iran is biased.
The Western implementation of democracy is an evolving compromise between elites and the masses. It is not perfect, for sure. Also, it is quite different across countries, and it is very different across the decades of the past century. When talking of "western democracy", we have to look for common elements.
It has settled on a few common ideas: one adult, one vote, freedom of speech, freedom of assembly, the rule of law (that is, equality before the law). The UK offers the supremacy of parliament. Systems with constitutions constrain the power of government, but through a mechanism subject to oversight by election, direct or indirect. Most of these constitutions were imposed by a process very different from the electoral processes the constitutions prescribe, which is an obvious contradiction. In a few cases, they were imposed by an occupying power, which could not be less democratic, you may say. I guess this is the root of the enduring hope that democracy can be spread by the US military.
Our democracies nearly always have means to suppress the direct will of the people, most obviously by delegation of authority to elected representatives which is a process subject to manipulation, but also through mechanisms like long terms of office for Sentators which delay rapid response to voter concerns, difficult processes to change the constitution, and rights which protect minorities, and voting systems which are often designed to filter out the diversity of voters (including providing small groups of voters with outsized representation, as happens in the Senates of the US and Australia). But despite that, Western systems can mostly claim that all power ultimately rests with the voters, even if the execution of this power is coralled though a maze set up by an older or external power structure.
So overall, it's pretty hard to use "western democracy" as a benchmark. And by today's standards, our systems of just 100 years ago looked pretty bad, and that's forgetting that most Western democracies are quite young: in 1940 there were only a few remaining. The "new world" countries are called "Western" due to their historical connections to Western Europe rather than their actual location, but they are actually much more mature democracies, and perhaps much better at it, than nearly all the countries of Western Europe. Of those still-standing democracies in 1940, Western Europe could muster only one important member. So historically, "Western Democracy" should be taken with a grain of salt. The legitimacy of today's "Western Democracy" comes from outside of Western Europe, except for the UK.
However, the Iranian system fails so many of the broad tests, it is not really a question of subtle differences. It is clearly in a different league. The state enforces brutal suppression of opponents, it limits candidates, it limits free speech and so on.
But this is also not the end of the discussion, I think. From the beginning of the theocratic takeover of a very brutal and old-fashioned oil-funded dictatorship (propped-up by the western powers), the religious institution has definitely sought legitimacy via elections. It is quite interesting. While claiming overall authority, from the beginning, the religious leaders recognised an important role for secular institutions. Of course eventually this caused conflicts and challenges to the constraints, and the secular institutions become a threat, and so far the institutional power of the elite, which has vast economic interests, not just Ayatollahs, is winning.
Iran has the kernel of a representative democracy. The right to vote is as I understand it wide-spread, and the theocratic regime has significantly improved literacy. It seems that at times, Iranian voters have felt like they were participating in a legitimate process. Also, it is not just at the national level: there are elections down to municipal level, which indicates a depth to the concept of elected representation. These proto-democratic institutions are certainly perceived as a threat by the entrenched elite, which is a sign that they have some independent life. There is definitely widespread political opposition to the regime. The regime has never succeeded in eliminating this opposition, there must be either be leadership dissent or some other limits on the type of brutality required to do this.
Iran is not so far away from producing a nuclear weapon. The "bomb" of democracy may not be so far either.
Sometimes the "West" embarks on ridiculously futile campaigns, such as the attempt to establish democratic institutions in Afghanistan or Iraq.
Iran seems like a much more likely prospect in many ways. It is not democratic, but there is a democratic Iran not too far away if things go well.