Does a strong dollar achieve the same tightening effect on the economy as rising interest rates? Rising interest rates means the cost of borrowing USD goes up. A stronger USD means USD becomes more expensive. Wouldn't that naturally follows that the cost of borrowing USD goes up since each dollar becomes more expensive?

Does strong USD achieve the same effect as rising interest rates? If yes, is it logical to conclude that the Fed would put off raising interest rates if the dollar rises too much?
ceteris paribus, you are absolutely right. The price for international borrowing and lending is determined by both the interest and exchange rate.

Three remarks:


Above; I wrote ceteris paribus, meaning nothing else changes. In reality, however, both (long-term) interest rates and exchange rates are heavily impacted by expectations on how the economy is going to evolve.
Note that the exchange rate only impacts international transactions, on both financial and goods markets, whereas the interest rate impacts all borrowing and lending. The impact of a change in the exchange rate is therefore mainly a quantitative question. Without going into too much detail, the effect is likely going to be a lot bigger for a small open economy, compared to the US.
Usually, causation is assumed to go the other way. That is, the Federal reserve sets interest rates (arguably on the backdrop of domestic economic conditions), and then the exchange rate reacts on changes in interest rates. However, this is conventional wisdom, not an economic fact.