"We're just at the beginning of the commotion" in Rome, said the veteran strategist, referring to whether Italy might ultimately seek to leave the euro, causing a similar upheaval to when the U.K. voted to leave the overall European Union.
Uncertainty about the euro's future could push the dollar even higher, said Grant, who has been working in the financial industry for more than four decades.
"I think it could go to $1.14, maybe $1.12 [against the euro]," he said. "This could have a major impact on the equity markets in the United States in terms of earnings of our international companies."
A stronger dollar makes goods sold by U.S. multinational companies overseas more expensive, which can put them at a pricing disadvantage to their foreign competitors.
The U.S. currency strengthens when the number on the chart goes lower because it costs fewer dollars to buy euros.
Since hitting a 2018 low of about $1.25 per euro in mid-February, the dollar has gained 8 percent to about $1.15 per euro as of Tuesday's settling price.
The Dow Jones industrial average sank nearly 400 points in Tuesday's Italy-fueled global stock sell-off. U.S. stock futures in premarket Wednesday trading were pointing to a recovery of about a quarter of those losses.
At issue is Italy's struggle to form a new government and whether euroskeptic populists will secure an unprecedented governing mandate there, which could threaten the future of the euro zone's third-largest economy in the common currency bloc.
Grant sees two possible scenarios that are worrying the investment community: Italy could vote to leave the euro; or Italy could refuse to abide by European Union budgetary demands and get kicked out of the euro.
"One of the real dangers here, if for any reason that Italy leaves and re-denominates its currency, the ECB would take a tremendous hit that could literally bankrupt them unless the other countries put up more money."