Viability of Bitcoin due to transaction fee? I've been thinking about this for quite some time, and spent some time searching for information online, but haven't really found good answers. 

The transaction fees of crypto currencies are relatively low for large transfers, but high for small transfers. As the price of bitcoin rises, the transaction cost also rises. Won't this cause issues in the long run for normal day-to-day use of Bitcoin, resulting in it only being used for (fairly) large transfers or money?

According to https://bitcoinfees.21.co/: 


  The fastest and cheapest transaction fee is currently 360 satoshis/byte, shown in green at the top.
  For the median transaction size of 226 bytes, this results in a fee of 81,360 satoshis.


Currently, that price translates to EUR 2.07001 or USD 2.31862, according to https://www.xe.com. If you're sending e.g. 1000 EUR, then that amount is fine, but if you're sending 10 EUR (I'm imagining a Bitcoin ATM), the transaction fees will be way to high.

To me it seems these fees will be a huge impediment on the acceptance of the currency by "civilians"?
Relative economic advantage

As you've noted, transaction fees are paid for block space and not relative to the amount that is transferred. This causes an obvious economic advantage to larger value transfers and would inevitably push out smaller transfers if supply remains the same and demand increases.

Supply increase

This is why for several years, there have been proposals to increase the blockspace supply. This would probably mitigate the problem in the short term, however would probably quickly reestablish the same problem at a lower fee level that would continue to rise as demand catches up.

There have been some proposals to create automatically adapting blocksizes, but these proposals have so far been incomplete and especially deficient in providing a mechanism to push for blocksize decreases. The latter is important to actually achieve a self-balancing system as a quickly growing blocksize would cause other parts of the system to fail due to vastly driving up the cost-of-node-option.

Demand management

The other side of increasing the supply would be to manage the demand. This has happened in the past to some degree by transactions moving offchain, e.g. into third party custodial wallets such as the service ChangeTip that facilitated tipping internet contributions or wallets such as Coinbase. A lot of work has gone into a decentralized second layer that basically acts as a aggregation layer for transactions. This Lightning Network is thought to settle on a fee system relative to the transferred value as the costs of relaying payments scale with the amount transferred. If it should gain widespread use it is expected to provide a haven for small value transactions that automatically rebalances with on-chain demand.