These are anxious times for Mario Draghi. The giant government bond-buying program that the European Central Bank president launched last year has achieved a great deal. It has driven down borrowing costs for governments, companies and banks; it has lowered the exchange rate; it has inflated asset prices, easing the pressure on bank balance sheets. It enabled the eurozone to survive the Greek crisis last year and the potentially destabilizing impact of inconclusive elections in Portugal, Spain, Ireland and Slovakia.

The only thing it hasn’t done is the one thing it was supposed to do: to bring inflation back to its target of close to but below 2%.

Instead, headline inflation in the eurozone last month fell to minus 0.2% while core inflation, which strips out the effects of volatile energy prices, unexpectedly fell to 0.7%, down from 1% in January.

If the eurozone was a normal economy, this might not matter so much. Growth and inflation may be currently running below previous forecasts, but a normal central bank might be willing to look through this temporary weakness. J.P. Morgan reckons that the ECB will have to cut its forecast for growth next year to 1.8% from 1.9% and core inflation to 1.4% from 1.6%, and to 1.7% for both metrics in 2018—more or less in line with the inflation target.

Besides, there is little evidence that current disinflation is feeding through to the kind of self-fulfilling deflationary spiral that keeps central bankers awake at night. Prices have been falling in Spain for two years, yet consumer spending keeps rising. The Bank for International Settlements, or BIS, has shown that with the exception of the Great Depression, falling prices have historically been associated with growth.

But the eurozone isn’t a normal economy, which is why the markets regard Thursday’s meeting of the ECB Governing Council as another make-or-break moment for the single currency. The problem is the very high level of public and private debt in some parts of the eurozone, which becomes much harder to tackle at very low levels of inflation.

Italy’s government debt of 134% of gross domestic product, for example, looks perfectly sustainable when inflation is 2%. That is the case even if growth is only 1%, since government-spending freezes should ensure that the budget deficit shrinks quickly, allowing the debt-to-GDP ratio to fall. But if inflation were to continue to flatline well below target, Italy’s debt burden would quickly start to look more challenging. The ECB is therefore under pressure to convince the markets that it has the ability and will to return inflation quickly to target.

Indeed, Mr. Draghi has bought some of this pressure on himself. He has warned that if the ECB doesn’t act decisively to tackle low inflation, the markets will start to doubt its commitment to its inflation target, undermining inflation expectations. The result is that investors will dial down their expectations of future nominal returns, leading to a negative spiral of lower investment and lower growth.

Moreover, he has insisted that the ECB does have the tools to bring inflation back to target and isn’t afraid to use them. In December, the markets judged that he failed to live up to his rhetoric. Since then, he has hardened his rhetoric while taking care to avoid discussing specific policies. If he disappoints markets a second time, he risks triggering the very spiral he has warned about, as investors will conclude his rhetoric is empty.

The reality is that the ECB is running out of tools to fight low inflation—or at least running out of tools that are easy to activate and which are likely to have a sizable impact. Investors are increasingly concerned that negative interest rates—a preferred ECB policy until now—may be doing more harm than good, a view echoed this week by the BIS.

What the market really craves now is a signal that the ECB is willing to expand its quantitative easing far beyond its current, self-imposed limits. That could involve buying bonds at prices that guarantee a loss, or allowing itself to acquire larger proportions of individual bond issues, potentially giving it a controlling stakes in the event of a debt restructuring, or abandoning its policy of buying bonds strictly in proportion to each country’s ECB shareholding. But each of these are politically difficult, exposing the ECB to accusations that it is breaching the treaty prohibition on financing governments.

Mr. Draghi may be able to conjure together a package that satisfies the markets for now, delivering enough stimulus to push up near-term inflation by pushing down the euro and importing some demand from the rest of the world. But what he can’t do is force businesses to invest. Only governments can take the steps to remove the political clouds that are holding back the eurozone’s animal spirits and thereby lift long-term growth and productivity.

If the European Union’s Schengen passport-free travel zone collapses, if Britain exits the EU, if Italy and Portugal fail to tackle their mountains of bad debt, if France can’t overhaul its labor market, if populists opposed to reforms continue to win elections, if eurozone countries are unable to agree on measures to boost the resilience of the currency bloc, then Mr. Draghi’s most anxious days may lie ahead.